Download circle - einstein classes

MC – 1
CIRCLE
A circle is the locus of a point moving in a plane so that it always remains at a constant distance from a fixed
point. The fixed point is called its center and the constant distance is called its radius.
C1
Equation of a Circle in different forms :
Forms
Equation
Standard Form
(x – h)2 + (y – k)2 = r2
Figure
here (h, k) is the centre and
r is the radius
Centre at the origin
x2 + y2 = r2
Circle passes through
the origin
x2 + y2 – 2hx – 2ky = 0
Circle touches x-axis
x2 + y2 – 2hx – 2ay + h2 = 0
Circle touches y-axis
x2 + y2 – 2ax – 2ky + k2 = 0
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 2
Forms
Equation
Figure
Circle touches both the x2 + y2 – 2ax – 2ay + a2 = 0
axes
Diameter Form
General Form
(x – x1) (x – x2) + (y – y1)
(y – y2) = 0
x2 + y2 + 2gx + 2fy + c = 0 is the circle whose center is at (–g, –f)
and radius is
g2  f 2  c .
Notes
(i)
If g2 + f2 – c > 0, then the circle is
called a real circle.
(ii)
If g2 + f2 – c = 0, then the circle is
called a point circle.
(iii)
If g2 + f2 – c < 0, then the circle is
called an imaginary circle.
(iv)
The lengths of intercepts made by
the circle x2 + y2 + 2gx + 2fy + c = 0
with x and y axes are
:
2 g 2  c and 2 f 2  c
respectively.
Parametric form
Einstein Classes,
The parametric equations of (x – h)2 + (y – k)2 = r2 and :
x = h + r cos  ; y = k + r sin  ; –  <    where (h, k) is the
centre, r is the radius &  is a parameter
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 3
Practice Problems :
1.
Prove that the centres of the circles x2 + y2 = 1, x2 + y2 + 6x – 2y – 1 = 0 and x2 + y2 – 12x + 4y = 1 are
collinear.
2.
If the equation of two diameters of a circle are 2x + y = 6 and 3x + 2y = 4 and the radius is 10, find the
equation of the circle.
3.
Find the equation of the circle whose centre is (1, 2) and which passes through the point of
intersection of 3x + y = 14 and 2x + 5y = 18.
4.
Find the equation of the circle whose radius is 5 and the centre lies on the positive sides of x-axis at
a distance 5 from the origin.
5.
Find the equation of the circle which passes through the points (–1, 2) and (3, –2) and whose centre
lies on the line x – 2y = 0.
6.
Find the equation of the circle which touches both the axes and whose radius is a.
7.
Find the equation of the circle, the end point of whose diameter are (2, –3) and (–2, 4). Find its
centre and radius.
8.
Find the equation of the circle passing through the points (1, 0), (0, 1) and (1, –2).
[Answers : (2) x2 + y2 – 16x + 20y + 64 = 0 (3) x2 + y2 – 2x – 4y – 4 = 0 (4) x2 + y2 – 10x = 0
 1  65
(5) x2 + y2 – 4x – 2y – 5 = 0 (6) x2 + y2 ± 2ax ± 2ay – a2 = 0 (7) x2 + y2 – y – 16 = 0;  0,  ;
 2 2
2
2
(8) x + y + 2x + 2y – 3 = 0]
C2
Position of a point with respect to a circle :
A point (x1, y1) lies inside, on or outside the circle S  x2 + y2 + 2gx + 2fy + c = 0 according as S1 < 0,
S1 = 0 or S1 > 0 respectively, where S1  x12 + y12 + 2gx1 + 2fy1 + c.
NOTE : The greatest and the least distance of a point A from a circle with centre C and radius r is AC + r
and AC – r respectively.
Practice Problems :
1.
Does the point (–2.5, 3.5) lie inside, outside or on the circle x2 + y2 = 25 ?
2.
Discuss the position of the points (1, 2) and (6, 0) with respect to the circle x2 + y2 – 4x + 2y – 11 = 0.
3.
Find the minimum and maximum distance from the point (2, –7) to the circle
x2 + y2 – 14x – 10y – 151 = 0
4.
If the point (, –) lies inside the circle x2 + y2 – 4x + 2y – 8 = 0, then find range of .
[Answers : (2) (1, 2) lies inside and (6, 0) lies outside (3) minimum : 2, maximum : 28 (4)   (–1, 4)]
C3
Straight Line and a Circle :
Let L = 0 be a line and S = 0 be a circle. If r is the radius of the circle and p is the length of the perpendicular
from the centre on the line, then
(i)
p>r

the line does not meet the circle i.e. passes out side the circle.
(ii)
p=r

the line touches the circle. (It is tangent to the circle)
(iii)
p<r

the line is a sacant of the circle
(iv)
p=0

the line is a diameter of the circle
Note the following points :
1.
The line y = mx + c intersects circle x2 + y2 = a2 in two distinct points if c2 < a2 (1 + m2).
2.
The line y = mx + c touches circle x2 + y2 = a2 if c2 = a2(1 + m2) at point
Einstein Classes,
  ma 2 a 2 

,  .
c 
 c
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 4
3.
The line y = mx + c does not intersect the circle x2 + y2 = a2 at all if c2 > a2 (1 + m2).
4.
Equation of the pair of straight lines passing through the origin and the points of intersection of the line
lx + my + n = 0 and the circle x 2 + y 2 + 2gx + 2fy + c = 0 is x 2 + y 2 + (2gx + 2fy)
2
lx  my
 lx  my 
 c
  0.
n
 n 
Practice Problems :
1.
Prove that for all values of , x sin  – y cos  = p touches the circle x2 + y2 = p2.
2.
Find the equation of the circle whose centre is (1, –3) and which touches the line 2x – y – 4 = 0.
3.
Write down the equation of a circle concentric with the circle x2 + y2 – 4x + 6y – 17 = 0 and tangent to
the line 3x – 4y + 7 = 0.
4.
If the line px + qy + r = 0 touches the circle x2 + y2 = a2 then prove that r2 = a2(p2 + q2).
5.
Find those tangents to the circle x2 + y2 = 16 which are parallel to 3x – 16y = 10.
6.
Show that the line 7y – x = 5 touches the circle x2 + y2 – 5x + 5y = 0 and find the equation of the other
parallel tangent.
[Answers : (2) 5(x2 + y2) – 10x + 30y + 49 = 0 (3) x2 + y2 – 4x + 6y – 12 = 0 (5) 3x – 16y ± 4265 = 0
(6) x – 7y – 45 = 0]
C4
The length of the intercept cut off from a line by a circle :
The intercepts made by the circle x2 + y2 + 2gx + 2fy + c = 0 on the co-ordinates axes are 2 g2  c and f 2  c
respectively. If
g2 > c

circle cuts the x axis at two distinct points
2

circle touches the x axis
2
g <c

circle lies completely above or below the x-axis
f2 > c
g =c
Similarly,

circle cuts the y-axis at two distinct points
2

circle touches the y-axis
2

circle lies completely above or below the y-axis
f =c
f <c
The length of the intercept cut off from the line y = mx + c by the circle x2 + y2 = a2 is
2
a 2 (1  m 2 )  c 2
(1  m 2 )
.
Condition of Tangency : The line y = mx + c touches the circle x2 + y2 = a2 if the length of the intercept is
zero. i.e. c = ± a 1  m 2 .
Practice Problems :
1.
Find the value of  so that the line 3x – 4y = , may touch the circle x2 + y2 – 4x – 8y – 5 = 0.
2.
Find the length of the intercept on the straight line 4x – 3y – 10 = 0 by the circle
x2 + y2 – 2x + 4y – 20 = 0.
3.
Find the coordinates of the middle point of the chord which the circle x2 + y2 + 4x – 2y – 3 = 0 cuts off
the line x – y + 2 = 0.
 3 1
[Answers : (1) 15, –35 (2) 10 (3)   ,  ]
 2 2
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 5
C5
Tangent to a Circle :
(i)
Slope Form : Equations of tangents to the circle x2 + y2 = r2 in slope form are
(a)
  mr
,
y  mx  r 1  m 2 . It touches the circle at the point 

(b)
y  mx  r 1  m 2 . It touches the circle at the point 
(ii)
Point Form :
(a)
Equation of the tangent to the circle S  x2 + y2 + 2gx + 2fy + c = 0 at the point (x1, y1) is
T  xx1 + yy1 + g(x + x1) + f (y + y1) + c = 0.
(b)
The equation of the tangent to the circle x2 + y2 = r2 if c2 = r2 (1 + m2). Hence equation of tangent
2
 1 m

mr
2
 1 m
,

.

1 m2 
r

.
2 
1 m 
r
 r 2m r 2 
is y  mx  r 1  m 2 and the point of contact is  
, .

c
c 

Note : In general the equation of tangent to any second degree curve at point (x1, y1) on it can be obtained
by replacing x2 by x x1, y2 by yy1, x by
(iii)
x y  xy1
x  x1
y  y1
, xy by 1
and c remains as c.
, y by
2
2
2
Parametric Form : The equation of a tangent to circle x2 + y2 = r2 at (r cos , r sin ) is
x cos  + y sin  = r.
Practice Problems :
1.
Prove that the tangents to the circle x2 + y2 = 25 at (3, 4) and (4, –3) are perpendicular to each other.
2.
Find the equation of tangent to the circle x2 + y2 – 2ax = 0 at the point [a(1 + cos ), a sin ]
3.
Find the equations of the tangents to the circle x2 + y2 = 9, which
(i)
are parallel to the line 3x + 4y – 5 = 0
(ii)
are perpendicular to the line 2x + 3y + 7 = 0
(iii)
make an angle of 600 with the x-axis.
4.
Prove that the line lx + my + n = 0 touches the circle (x – a)2 + (y – b)2 = r2 if (al + bm + n)2 = r2(l2 + m2).
5.
Show that the line 3x – 4y = 1 touches the circles x2 + y2 – 2x + 4y + 1 = 0. Find the co-ordinates of the
point of contact.
6.
Show that the line (x – 2) cos  + (y – 2) sin  = 1 touches a circle for any values of . Find the circle.
[Answers : (2) x cos  + y sin  = a(1 + cos ) (3) (i) 3x + 4y ± 15 = 0 (ii) 3x – 2y ± 313 = 0
(iii) 3x – y ± 6 = 0 (6) x2 + y2 – 4x – 4y + 7 = 0]
C6
Normal : If a line is normal/orthogonal to a circle then it must pass through the centre of the circle. Using
this fact normal to the circle x2 + y2 + 2gx + 2fy + c = 0 at (x1, y1) is y  y 1 
y1  f
( x  x1 ) .
x1  g
Practice Problems :
1.
Find the equation of the normal to the circle x2 + y2 = 2x, which is parallel to the line x + 2y = 3.
2.
Find the equation of the normal to the circle x2 + y2 – 5x + 2y – 48 = 0 at the point (5, 6).
3.
If the radius of the circle is 5 and the equations of the two normals to the circle are 3x – 5y + 2 = 0 and
x + 2y = 3, find the equation of the circle.
[Answers : (1) x + 2y – 1 = 0 (2) 14x – 5y – 40 = 0 (3) x2 + y2 – 2x – 2y – 23 = 0]
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 6
C7
Length of Tangent : The length of the tangent from the point P(x 1 , y 1 ) to the circle
x2 + y2 + 2gx + 2fy + c = 0 is equal to
x12  y12  2gx 1  2fy1  c  S1
Note : If PQ is a length of the tangent from a point P to a given circle, then PQ2 is called the power of the
point with respect to a given circle.
Practice Problems :
1.
Find the length of tangents drawn from the point (3, –4) to the circle 2x2 + 2y2 – 7x – 9y – 13 = 0.
2.
If the length of the tangents from (f, g) to the circle x2 + y2 = 6 be twice the length of the tangent from
(f, g) to circle x2 + y2 + 3x + 3y = 0 then will f2 + g2 + 4f + 4g + 2 = 0 ?
3.
Show that the area of the triangle formed by the tangents from the point (4, 3) to the circle
x2 + y2 = 9 and the line segment joining their points of contact is 7
4.
17
square unit in length.
25
Show that the length of the tangent from any point on the circle x2 + y2 + 2gx + 2fy + c = 0 to the circle
x2 + y2 + 2gx + 2fy + c1 = 0 is
(c 1  c ) .
5.
Find the power of point (2, 4) with respect to the circle x2 + y2 – 6x + 4y – 8 = 0.
6.
Show that the locus of the point, the powers of which with respect to two given circles are equal, is a
straight line.
[Answers : (1) 26 (2) yes]
C8
Combined Equation of Pair of Tangents : Combined equation of the pair of tangents to the circle S  x2
+ y2 + 2gx + 2fy + c = 0 drawn from the point (x1, y1) is T2 = SS1, where T and S1 have their usual meaning.
Practice Problems :
1.
Find the equations of the tangents to the circle x2 + y2 = 16 drawn from the point (1, 4).
2.
The angle between a pair of tangents from a point P to the circle
x2 + y2 + 4x – 6y + 9 sin2 + 13cos2 = 0 is 2. Find the equation of the locus of the point P.
3.
Find the equation of the tangents from the point A(3, 2) to the circle x2 + y2 + 4x + 6y + 8 = 0.
[Answers : (1) 8x + 15y = 68 (2) (x + 2)2 + (y – 3)2 = 4 (3) 2x – y – 4 = 0 and x – 2y + 1 = 0]
C9A Chord of Contact : If two tangents PT 1 & PT 2 are drawn from the point P(x1, y1) to the circle
S  x2 + y2 + 2gx + 2fy + c = 0, then the equation of the chord of contact T1T2 is :
xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0
Note :
Here R = radius; L = length of tangent.
(a)
Chord of contact exists only if the point ‘P’ is not inside.
(b)
Length of chord of contact T1T2 
Einstein Classes,
2LR
R 2  L2
.
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 7
C9B
RL3
(c)
Area of the triangle formed by the pair of the tangents & its chord of contact =
(d)
 2RL 

Tangent of the angle between the pair of tangents from (x1, y1) =  2
 L  R2 
(e)
Equation of the circle circumscribing the triangle PT1T2 is (x – x1) (x + g) + (y – y1) (y + f) = 0
R 2  L2
Equation of a Chord Joining Two Point On The Curve :
Let P(x1, y1) and Q (x2, y2) be a two point on a circle x2 + y2 + 2gx + 2fy + c = 0
Then the equation of chord joining any two point on the curve is given by
(x – x1) (x1 + x2 + 2g) + (y – y1) (y1 + y2 + 2f) = 0
The equation of the chord when P = (–g + r cos 1, – f + r sin 1) and Q = (–g + r cos 2, – f + r sin 2) are
on the circle x2 + y2 + 2gx + 2fy + c = 0 is given by
   2 
  
  
2
2
( x  g ) cos 1
  ( y  f ) sin  1 2   r cos 1 2  where r  g  f  c .
2
2
2






C9C Equation of a Chord in Terms of its Mid Point : Let (x1, y1) be the mid point of a chord of the circle
S  x2 + y2 + 2gx + 2fy + c = 0, then the equation of the chord is T = S1.
Practice Problems :
1.
Find the condition that chord of contact of any external point (h, k) to the circle x2 + y2 = a2 should
subtend right angle at the centre of the circle.
2.
The chord of contact of tangents drawn from a point on the circle x2 + y2 = a2 to the circle x2 + y2 = b2
touches the circle x2 + y2 = c2. Show that a, b, c are in G.P.
3.
Find the equation of the chord of x2 + y2 – 6x + 10y – 9 = 0 which is bisected at (–2, 4).
4.
Find the middle point of the chord intercepted on line lx + my + n = 0 by the circle x2 + y2 = a2.
5.
Through a fixed point (h, k), secants are drawn to the circle x2 + y2 = r2. Show that the locus of mid
point of the portions of secants intercepted by the circle is x2 + y2 = hx + ky.
6.
Find the locus of middle points of chords of the circle x2 + y2 = a2, which subtend right angle at the
point (c, 0).
[Answers : (1) h2 + k2 = 2a2 (3) 5x – 9y + 46 = 0 (6) 2(x2 + y2) – 2cx + c2 – a2 = 0]
C10
Director Circle : The locus of the point of intersection of two perpendicular tangents to a circle is called
the Director Circle.
Equation of director circle : Let the circle x2 + y2 = a2. Then equation of the pair of tangents to a circle
from a point (x1, y1) is (x2 + y2 – a2) (x12 + y12 – a2) = (xx1 + yy1 – a2)2. If this represent a pair of perpendicular
lines then.
The essential condition is coefficient of x2 + coefficient of y2 = 0
i.e.
(x12 + y12 – a2 – x12) + (x12 + y12 – a2 – y12) = 0

x12 + y12 = 2a2
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 8
Hence the equation of director circle is x2 + y2 = 2a2

Director circle is a concentric to the given circle but whose radius is 2 times the radius of the
given circle.
Practice Problems :
1.
If two tangents are drawn from a point on the circle x2 + y2 = 50 to the circle x2 + y2 = 25 then find the
angle between the tangents.
[Answers : (1) 900]
C11
Family of Circles :
(i)
Let S  x2 + y2 + 2gx + 2fy + c = 0 be a circle and L  ax + by + k = 0 be a line intersecting S = 0, then the
equation of the family of circles passing through the intersection of the given circle and the line is
S +  L = 0, where  is a parameter.
(ii)
Let S = 0 and S’ = 0 be two intersecting circles. Then equation of the family of circles passing through
points of intersection of S = 0 and S’ = 0 is S +  S’ = 0, where   – 1. S +  (S – S’) = 0 also represent a
family of circles through the point of intersection of the circles S = 0 and S’ = 0.
(iii)
Equation of the family of circles each member of which touches the line ax + by + c = 0 at a point (x1, y1)
is (x – x1)2 + (y – y1)2 +  (ax + by + c) = 0.
(iv)
Equation of the family of circles each member of which passes through the points (x1, y1) and (x2, y2) is (x
x
– x1) (x – x2) + (y – y1) (y – y2) +  x1
x2
y 1
y1 1  0 .
y2 1
(v)
Equation of the family of circles each member of which touches the x-axis is x2 + y2 + 2gx + 2ay + g2 = 0.
If a > 0, then the circles lie below x-axis.
(vi)
Equation of the family of circles each member of which touches the y-axis is x2 + y2 + 2ax + 2fy + f2 = 0. If
a < 0, then the circles lie on the right of y-axis.
(vii)
Equation of the family of circles each member of which touches the both axes and lies in the first quadrant
is x2 + y2 – 2ax – 2ay + a2 = 0, where a > 0.
(viii) Equation of the family of circles each member of which touches both the axes and lies in the second
quadrant is x2 + y2 + 2ax – 2ay + a2 = 0, where a > 0.
(ix)
Equation of the family of circles each member of which touches both the axes and lies in the third quadrant
is x2 + y2 + 2ax + 2ay + a2 = 0, where a > 0.
(x)
Equation of the family of circles each member of which touches both the axes and lies in the fourth
quadrant is x2 + y2 – 2ax + 2ay + a2 = 0, where a > 0.
Practice Problems :
1.
Find the equation of the circle passing through (1, 1) and the points of intersection of the circles
x2 + y2 + 13x – 3y = 0 and 2x2 + 2y2 + 4x – 7y – 25 = 0.
2.
Find the equation of the circle passing through the point of intersection of the circles
x2 + y2 – 6x + 2y + 4 = 0, x2 + y2 + 2x – 4y – 6 = 0 and with its centre on the line y = x.
3.
Find the equation of the circle passing through the points of intersection of the circles
x2 + y2 – 2x – 4y – 4 = 0 and x2 + y2 – 10x – 12y + 40 = 0 and whose radius is 4.
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 9
4.
Find the equation of the circle through points of intersection of the circle x2 + y2 – 2x – 4y + 4 = 0 and
the line x + 2y = 4 which touches the line x + 2y = 0.
[Answers : (1) 4x 2 + 4y 2 + 30x – 13y – 25 = 0 (2) 7x 2 + 7y 2 – 10x – 10y – 12 = 0
(3) 2x2 + 2y2 – 18x – 22y + 69 = 0 and x2 + y2 – 2y – 15 = 0 (4) x2 + y2 – x – 2y = 0]
C12
Two Circles : Let there be two circles with centers at C1 and C2 and radii r1 and r2 respectively, then
(i)
Two circles are exterior to each other if r1 + r2 < C1C2
(ii)
Two circles touch each other externally if r1 + r2 = C1C2
(iii)
Two circles intersect each other in two points if |r1 – r2| < C1C2 < r1 + r2
(iv)
Two circles touch each other internally if |r1 – r2| = C1C2
(v)
One circle is interior to the other if C1C2 < |r1 – r2|.
Practice Problems :
1.
Examine if the two circles x2 + y2 – 2x – 4y = 0 and x2 + y2 – 8y – 4 = 0 touch each other externally or
internally.
2.
Prove that the circles x2 + y2 + 2ax + c2 = 0 and x2 + y2 + 2by + c2 = 0 touch each other, if
1
a
2

1
b
2

1
c2
.
[Answers : (1) internally]
C13
Common Tangents to two Circles :
Case
(i)
Number of Tangents
Condition
4 common tangents
r1 + r2 < c1 c2
(2 direct and 2 transverse)
(ii)
3 common tangents
r1 + r2 = c1 c2
(iii)
2 common tangents
|r1 – r2| < c1 c2 < r1 + r2
(iv)
1 common tangent
|r1 – r2| = c1 c2
(v)
No common tangent
c1 c2 < |r1 – r2|
(Here c1c2 is distance between centres of two circles)
Notes :
(i)
The direct common tangents meet at a point which divides the line joining centre of circles
externally in the ratio of their radii.
(ii)
Length of an external (or direct) common tangent & internal (or transverse) common tangent to
the two circles are given by :
L ext  d 2  (r1  r2 ) 2 & L int  d 2  (r1  r2 ) 2 , where d = distance between the centres of the two
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 10
circles and r1, r2 are the radii of the two circles. Note that length of internal common tangent is always less
than the length of the external or direct common tangent.
Practice Problems :
1.
Find all the common tangents to the circle
x2 + y2 – 2x – 6y + 9 = 0
and
x2 + y2 + 6x – 2y + 1 = 0
2.
Show that the common tangents to the circles x2 + y2 – 6x = 0 and x2 + y2 + 2x = 0 form an equilateral
triangle.
C14
Radical Axis : Radical axis of two circles is the locus of a point from which tangents drawn to circles are
of equal lengths. Radical axis of circles S 1 = 0 and S 2 = 0 is the line S 1 – S 2 = 0 i.e.
2(g1– g2)x + 2 (f1 – f2) y + (c1 – c2) = 0.
The common point of intersection of the radical axes of three circles taken two at a time is called the radical
centre of three circles.
Notes :
(i)
If the two circles intersect each other, then their common chord is the radical axis.
(ii)
If the two circles touch each other, then their common tangent at their point of contact is the radical axis.
(iii)
Radical axis is always perpendicular to the line joining the centres of the two circles.
(iv)
Radical axis will pass through the mid point of the line joining the centres of the two circles only if the two
circles have equal radii.
(v)
Radical axis bisects a common tangent between the two circles.
(vi)
A system of circles, every two which have the same radical axis, is called a coaxal system.
(vii)
Pairs of circles which do not have radical axis are concentric.
Practice Problems :
1.
Prove that the length of the common chord of the two circles :
(x – a)2 + (y – b)2 = c2 and (x – b)2 + (y – a)2 = c2 is
4c 2  2(a  b ) 2 .
2.
Find the circle whose diameter is the common chord of the circles x2 + y2 + 2x + 3y + 1 = 0 and
x2 + y2 + 4x + 3y + 2 = 0.
3.
If two circles x2 + y2 + 2gx + 2fy = 0 and x2 + y2 + 2g x  2f y = 0 touch each other, then f g  fg .
4.
Show that the difference of the squares of the tangents to two coplanar circles from any point P in
the plane of the circles vaies as the perpendicular from P on their radical axis. Also prove that the
locus of a point such that the difference of the squares of the tangents from it to two given circles is
constant is a line parallel to their radical axis.
5.
Find the radical centre of circles x 2 + y 2 + 3x + 2y + 1 = 0, x 2 + y 2 – x + 6y + 5 = 0 and
x2 + y2 + 5x – 8y + 15 = 0. Also find the equation of the circle cutting them orthogonally.
6.
Find the radical centre of three circles described on the three sides 4x – 7y + 10 = 0, x + y – 5 = 0 and
7x + 4y – 15 = 0 of a triangle as diameters.
7.
Prove that the tangents from any point of a fixed circle of co-axial system to two other fixed circles of
the system are in a constant ratio.
[Answers : (2) 2x2 + 2y2 + 2x + 6y + 1 = 0 (3) gf   g f (5) x2 + y2 – 6x – 4y – 14 = 0 (6) (1, 2)]
C15
Angle of Intersection of Two Circles : Let  be the angle of intersection of two circles whose centers are
2
at C1 and C2 and their radii are r1 and r2 respectively, then cos (  – θ) 
2
2
r1  r2  C1C 2
.
2r1r2
Notes :
(i)
2
2
2
If  = 900, then the circles are said to be Orthogonal and then r1  r2  C 1C 2 .
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 11
(ii)
Circles x2 + y2 + 2gx + 2fy + c = 0 and
x 2  y 2  2gx  2f y  c  0 are orthogonal ifff
2gg  2ff   c  c .
Practice Problems :
1.
Find the angle between the circles
S : x2 + y2 – 4x + 6y + 11 = 0 and S  : x2 + y2 – 2x + 8y + 13 = 0.
2.
Show that the circles x2 + y2 – 6x + 4y + 4 = 0 and x2 + y2 + x + 4y + 1 = 0 are orthogonal to each other.
3.
Find the equation of the circle which cuts the circle x2 + y2 + 5x + 7y – 4 = 0 orthogonally, has its
centre on the line x = 2 and passes through the point (4, –1).
4.
Find the equation of the two circles which intersect the circles
x2 + y2 – 6y + 1 = 0 and x2 + y2 – 4y + 1 = 0, orthogonally and touch the line 3x + 4y + 5 = 0.
5.
Prove that the two circles, which pass through (0, a) and (0, –a) and touch the line y = mx + c, will cut
orthogonally, if c2 = a2(2 + m2).
6.
Find the equation of the circle which cuts orthogonally each of the three circles given below :
x2 + y2 – 2x + 3y – 7 = 0, x2 + y2 + 5x – 5y + 9 = 0 and x2 + y2 + 7x – 9x + 29 = 0.
[Answers : (1)  = 1350 (3) x2 + y2 – 4x + 2y + 1 = 0 (4) x2 + y2 – 1 and 4x2 + 4y2 – 15x – 4 = 0
(6) x2 + y2 – 16x – 18y – 4 = 0]
C16
Pole and Polar :
(i)
If through a point P in a plane of the circle there be drawn any straight line to meet the circle in
Q and R, the locus of the point of intersection of the tangents at Q & R is called the Polar of the
point P; also P is called the Pole of the Polar.
(ii)
The equation to the polar of a point P(x1, y1) w.r.t. the circle x2 + y2 = a2 is given
by xx1 + yy1 = a2, & if the circle is general then the equation of the polar becomes
xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0 i.e. T = 0. Note that if the point (x1, y1) be on the circle then
the tangent & polar will be represented by the same equation. Similarly if the point (x1, y1) be
outside the circle then the chord of contact & polar will be represented by the same equation.
(iii)
 Aa 2
Ba 2
Pole of a given line Ax + By + C = 0 w.r.t. circle x2 + y2 = a2 is  
,

 C
C

(iv)
If the polar of a point P pass through a point Q then the polar of Q passes through P.
(v)
Two lines L1 & L2 are conjugate of each other if Pole of L1 lies on L2 & vice versa. Similarly two
points P & Q are said to be conjugate of each other if the polar of P passes through Q &
vice-versa.

.


Practice Problems :
1.
Find the locus of the pole of the line lx + my + n = 0 with respect to the circle which touches y-axis at
the origin.
2.
The pole of a straight line with respect to the circle x2 + y2 = a2 lies on the circle x2 + y2 = 9a2. Prove
that the straight line touches the circle x2 + y2 = a2/9.
3.
Prove that the polar of a given point with respect to any one of the circles x2 + y2 – 2kx + c2 = 0, where
k is a variable, always passes through a fixed point, whatever be the value of k.
4.
Show that the polars of the point (1, –2) with respect to the circles x2 + y2 + 6x + 5 = 0 and
x2 + y2 + 2x + 8y + 5 = 0 coincide. Prove also that there is another point, the polars of which with
respect to these circles are the same and find its co-ordiantes.
[Answers : (1) y(lx – n) = mx2]
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 12
C17
SOME USEFUL HINTS & TIPS :
1.
If a square whose sides are parallel to the axes is inscribed in a circle, then
(i)
x-coordinates of the center is not equal to x-coordinate of any of its vertices.
(ii)
y-coordinates of the center is not equal to y-coordinate of any of its vertices.
2.
Coordinates of vertices of the square whose sides are parallel to y = x and y = –x, inscribed in a circle with
centre at (x0, y0) and radius r are (x0, y0 ± r) and (x0 ± r, y0).
3.
Let a circle passes through center of another circle and both touch each other, then radius of the inner circle
is half the radius of the other, further center of the inner circle lies on the join of the center of the bigger
circle and their point of contact.
4.
The locus of mid point on chords of a circle subtending a constant angle at the centre of the circle is always
a circle concentric to the given circle.
5.
Radius of the circle with centre at P, drawn orthogonal to a given circle is the length of tangent drawn from
P to the given circle.
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 13
SINGLE CORRECT CHOICE TYPE
1.
2.
3.
4.
If a chord of a circle x2 + y2 = 8 makes equal intercepts of length ‘a’ on the co-ordinates axes, then
(a)
|a| < 8
(b)
|a| < 42
(c)
|a| < 4
(d)
|a| > 4
The equation of the tangent to the circle x2 + y2 = a2
which makes with the axes a triangle of area a2 is
(a)
x ± y = 2a
(b)
x ± y = 2 a
(c)
x + y = 2a2
(d)
none of these
(a)
x2 + y2 = k2
(b)
x2 + y2 = 2k2
(c)
x2 + y2 = 3k2
(d)
none of these
9.
The tangents to x2 + y2 = a2 having inclination 
and  intersect at P. If cot  + cot  = 0, then the
locus of P is
(c)
x+y=0
(b)
xy=0
(d)
x–y=0
none of these
10.
The circles whose equations are x2 + y2 + c2 = 2ax
and x2 + y2 + c2 – 2by = 0 will touch one another
externally if
The circle x2 + y 2 – 6x – 10y + c = 0 does not
intersect or touch both axes of co-ordinates and the
point (1, 4) lies inside the circle. Then range of
possible values of c is given by
(a)
25 < c < 29
(b)
c > 29
(c)
c > 25
(d)
c < 25
If the chord of contact of tangents drawn from a
point on the circle x 2 + y 2 = a 2 to the circle
x2 + y2 = b2 touches the circle x2 + y2 = c2, then
a, b, c are in
(a)
A.P.
(b)
(c)
H.P.
(d)
x+y=2
(b)
x2 + y2 = 1
(c)
x2 + y2 = 2
(d)
x+y=1
If the circle C1 : x2 + y2 = 16 intersects another circle
C2 of radius 5 in such a manner that the common
chord is of maximum length and has a slope equal
to 3/4, the co-ordinates of the centre of C2 are
(a)
 9 12   9 12 
  , ;  , 
 5 5  5 5 
(b)
1 1
1
 2  2
2
c
a
b
(b)
 9 12   9 12 
  , ;  , 
 5 5  5 5 
(c)
1
1
1
 2  2
2
a
b
c
(c)
 12 9   12 9 
 , ;   , 
 5 5  5 5
(d)
none of the above
none of these
11.
none of these
(a)
1 1
1
 2  2
2
b
c
a
The distance between the chords of the tangents to
the circle x2 + y2 + 2gx + 2fy + c = 0 from the origin
and the point (g, f) is
G.P.
The locus of the mid point of a chord of the circle
x2 + y2 = 4, which subtends a right angle at the
origin is
(a)
(d)
6.
8.
If a circle of constant radius 3k passes through the
origin and meets the axes at A and B, the locus of
the centroid of OAB is
(a)
5.
7.
If two circles a(x 2 + y 2 ) + bx + cy = 0 and
A(x2 + y2) + Bx + Cy = 0 touch each other, then
(a)
g 2 + f2
(a)
aC = cA
(b)
bC = cB
1 2
(g  f 2  c)
2
(c)
aB = bA
(d)
aA = bB = cC
(b)
(c)
1 g2  f 2  c
.
2 g2  f 2
(d)
1 g2  f 2  c
.
2 g2  f 2
Einstein Classes,
12.
The point of intersection of the common chords of
three circles described on the three sides of a
triangle as diameter is
(a)
centroid of the triangle
(b)
orthocentre of the triangle
(c)
circumcentre of the triangle
(d)
incentre of the triangle
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 14
13.
14.
15.
16.
17.
Let PQ and RS be tangents at the extremities of the
diameter PR of a circle of radius r. If PS and RQ
intersect at a point X on the circumference of the
circle, then 2r equals
(a)
PQ.RS
(c)
2PQ.RS
PQ  RS
(b)
PQ  RS
2
2
PQ  RS
2
(d)
a parabola
(b)
a circle
(c)
an ellipse
(d)
a pair of straight lines
2
20.



, cos , cos are in
2
2
2
(a)
A.P.
(b)
(c)
H.P.
(d)
G.P.
none of these
The triangle PQR is inscribed in the circle
x2 + y2 = 25. If Q and R have co-ordinates (3, 4) and
(–4, 3) respectively, then Q P R is equal to
(a)
/2
(b)
/3
(c)
/4
(d)
/6
The number of common tangents to the circles
x2 + y2 = 4 and x2 + y2 – 6x – 8y = 24 is
(a)
0
(b)
1
(c)
3
(d)
4
Let P Q R be a right angled triangle, right angled
at P (2, 1). If the equation of the line QR is
2x + y = 3, then the equation of the pair of lines PQ
and PR is
(a)
3x2 – 3y2 + 8xy + 20x + 10y + 25 = 0
(b)
3x2 – 3y2 + 8xy – 20x – 10y + 25 = 0
(c)
3x2 – 3y2 + 8xy + 10x + 15y + 20 = 0
(d)
3x2 – 3y2 – 8xy – 10x – 15y – 20 = 0
If

1 
 m i ,  , i = 1, 2, 3, 4 are concyclic points, then
mi 

the value of m1m2m3m4 is
18.
If ,  and  are the parametric angles of three
points P, Q and R respectively, on the circle
x2 + y2 = 1 and A is the point (–1, 0). If the length of
the chords AP, AQ, AR are in G.P., then
cos
Let AB be a chord of the circle x 2 + y 2 = r 2
subtending a right angle at the centre. Then the
locus of the centroid of the triangle PAB as P moves
on the circle is
(a)
19.
(a)
1
(b)
(c)
0
(d)
–1
none of these
The co-ordinates of two points P and Q are (x1, y1)
and (x2, y2) respectively and O is the origin. If circles
be described on OP and OQ as diameters then
length of their common chord is
(a)
| x1 y 2  x 2 y 1 |
PQ
(c)
| x1 x 2  y 1 y 2 |
PQ
Einstein Classes,
(b)
| x1 y 2  x 2 y 1 |
PQ
(d)
| x1 x 2  y 1 y 2 |
PQ
ANSWERS (SINGLE CORRECT
CHOICE TYPE)
1.
c
11.
b
2.
b
12.
b
3.
d
13.
a
4.
c
14.
b
5.
c
15.
b
6.
d
16.
b
7.
a
17.
a
8.
b
18.
b
9.
c
19.
b
10.
a
20.
c
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 15
EXCERCISE BASED ON NEW PATTERN
1.
2.
3.
COMPREHENSION TYPE
Comprehension-1
The director circle of a circle is the locus of points
from which tangents to the circle are at right angles.
C1, C2, C3,.... is a sequence of circles such that Cr + 1
is the director circle of Cr. Let the equation of C1 be
x2 + y2 = 4.
The length of a chord of C11 which touches C10 is
(a)
642
(b)
64
(c)
322
(d)
32
The distance of the chord of contact of tangents of
a point on C11 with respect to C10 from the centre of
C10 is
(a)
642
(b)
322
(c)
64
(d)
32
The locus of the poles of the tangents to C9 with
respect to C10 is
(a)
x2 + y2 = 2 × 322 (b)
x2 + y2 = 322
2
2
2
(c)
x + y = 64
(d)
x2 + y2 = 2 × 162
Comprehension-2
Let C1 and C2 be two circles with radii r1 and r2
(r1 > r2). Let the distance between their centres be d
then
(i)
If d < |r1 – r2| then C2 is fully contained in
C1 and number of common tangents = 0.
(ii)
If d = r1 – r2 then C2 touches C1 and lies
in the interior of C1. The number of
common tangents must be one.
(iii)
r1 – r2 < d < r1 + r2 then C1 and C2 cut at
two real points. The number of common
tangents must be two and their
length =
(iv)
5.
6.
7.
8.
9.
10.
d 2  (r1  r2 ) 2 .
d 2  (r1  r2 ) 2 (the length of
11.
transverse common tangents = 0).
If d > r1 + r2 then C1 and C2 have four
common tangents. Length of direct
common tangent =
d 2  (r1  r2 ) 2 .
Length of transverse common tangent =
d 2  (r1  r2 ) 2
The common tangents of two circles (or two conices)
can be easily found. If y = mx + c be the common
tangent then the quadratic equation formed by
putting y = mx + c in the equation of the circle
(conic) must have equal roots.
Einstein Classes,
The number of common tangents to the circles
x2 + y2 – 2x – 6y + 9 = 0, x2 + y2 + 6x – 2y + 1 = 0 must
be
(a)
1
(b)
2
(c)
3
(d)
4
The length of the transverse common tangents to
the circle given in Q. 4 must be
(a)
2
(b)
4
(c)
2
(d)
none of these
The horizontal common tangents of the two circles
in Q. 4 must be
(a)
y=1
(b)
y=2
(c)
y=3
(d)
y=4
The common tangent of the circles in Q. 4 having
negative slope must be
(a)
3x + 4y = 10
(b)
2y = –5x
(c)
y = –x
(d)
None of these
The equation of the common tangents to the circle
a2
and parabola y2 = 4ax must be
2
(a)
y = x + 2a
(b)
y = x + 4a
(c)
y=x+a
(d)
None of these
Comprehension-3
Let the mirror image of the point A(5, 6) with
respect to the line 2x + 3y = 15 be B.
The co-ordinate of point B is
(a)
(2, 0)
(b)
(0, 2)
(c)
(1, 0)
(d)
(0, 1)
The equation of the circle described on AB as
diameter is
(a)
x2 + y2 – 6x – 6y + 5 = 0
(b)
x2 + y2 – 12x – 12y + 5 = 0
(c)
x2 + y2 – 12x – 6y + 5 = 0
(d)
x2 + y2 – 6x – 12y + 5 = 0
AC is any chord of the circle meeting the x-axis at
D such that AD = 10DC. The number of such chords
are
(a)
1
(b)
2
(c)
3
(d)
none
Comprehension-4
Acoording to the law of reflection
(a)
the incident ray, normal and reflected
ray are in the same plane
(b)
angle of incidence equals to angle of
reflection
For the following problems use the above law :
A ray of light incident at the point (3, 1) gets
reflected from the tangent at (0, 1) to the circle
x2 + y2 = 1. The reflected ray touches the circle. The
equation of the line along which the incident ray
moves is
x2  y 2 
If d = r1 + r2 and C1 and C2 touch
externally. The number of common
tangents must be three and their
length =
(v)
4.
12.
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 16
13.
(a)
3x + 4y – 13 = 0 (b)
4x – 3y – 13 = 0
(c)
3x – 4y + 13 = 0 (d)
4x – 3y – 10 = 0
A ball moving around the circle
x2 + y2 – 2x – 4y – 20 = 0 in anti-clockwise direction
leaves it tangentially at the point P(–2, –2). After
getting reflected from a straight line L it passes
through the centre of the circle. The perpendicular
5
on the straight line L.
2
The angle between the incident line of the ball and
reflected line of the ball is
(a)
300
(b)
450
(c)
600
(d)
none
The slope of the line L is
(B)
(C)
distance from P is
14.
43 3
(a)
34 3
(D)
23 3
(b)
34 3
23 3
(c)
(A)
(B)
(C)
(D)
(A)
44 3
(d)
none
MATRIX-MATCH TYPE
Matching-1
Column - A
Column - B
The number of integral (P)
1
values of  for which
x2 + y2 + x + (1 – )y + 5 = 0
is the equation of a circle
whose radius cannot
exceed 5, is
If a circle passes through (Q)
16
the point of intersection of
the lines 2x – y + 1 = 0 and
x + y – 3 = 0 with the axes
of reference then the value
of  is
The length of the chord of (R)
2
the circle
x2 + y2 + 4x – 7y + 12 = 0
along the y-axis is
The maximum number
(S)
–2
of points with rational
coordinates on a circle
whose centre is (3, 0) is
Matching-2
Column - A
Column - B
If the point A(1, 4) and B (P)
are symmetrical about
the tangent to the circle
x2 + y2 – x + y = 0 at the
origin then coordinates
of B are
Einstein Classes,
1.
2.
3.
4.
4
8
6, 

5
5
5.
The chords of contact of (Q)
(1/2, 1/2)
the pair of tangents to the
circle x2 + y2 = 1 drawn
from any point on the line
2x + y = 4 pass through the
point
The common chord of the (R)
(1/2, 1/4)
circle
x2 + y2 + 6x + 8y – 7 = 0
and a circle passing through
the origin, and touching the
line y = x, always passes
through the point
A tangent to the circle
(S)
(4, 1)
x2 + y2 = 1 through the
point (0, 5) cuts the circle
x2 + y2 = 4
at A and B.
The tangents to the circle
x2 + y2 = 4 at A and B meet
at C. The coordinates of C are
MULTIPLE CORRECT CHOICE TYPE
The point (1, 4) is inside the circle S whose
equation is of the form x2 + y2 – 6x – 10y + k = 0, k
being an arbitrary constant. The possible values of
k if the circle S neither touches the axes nor cuts
them
(a)
26
(b)
27
(c)
28
(d)
29
Let L1 be a straight line passing through the origin
and L 2 be the straight line x + y = 1. If the
intercepts made by the circle x2 + y2 – x + 3y = 0 on
L1 and L2 are equal then which of the following
equations can represent L1 ?
(a)
x+y=0
(b)
x–y=0
(c)
x + 7y = 0
(d)
x – 7y = 0
A point P(3, 1) moves on the circle x2 + y2 = 4 and
after covering a quarter of the circle leaves it
tangentially. The equation of a line along which the
point moves after leaving the circle is
(a)
y = 3x + 4
(b)
3y = x + 4
(c)
3y = x – 4
(d)
y = 3x – 4
Let the equation of a circle be x 2 + y 2 = a 2 .
If h2 + k2 – a2 < 0 then the line hx + ky = a2 is the
(a)
polar line of the point (h, k) with respect
to the circle
(b)
real chord of contact of the tangents from
(h, k) to the circle
(c)
equation of a tangent to the circle from
the point (h, k)
(d)
none of these
The equation of a circle C1 is x2 + y2 = 4. The locus
of the intersection of orthogonal tangents to the
circle in the curve C 2 and the locus of the
intersection of perpendicular tangents to the curve
C2 is the curve C3. Then
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 17
(a)
(b)
(c)
6.
7.
8.
C3 is a circle
the area enclosed by the curve C3 is 8
C2 and C3 are circles with the same
centre
(d)
none of these
Let a line through the point P(5, 10) cut the line l
whose equation is x + 2y = 5, at Q and the circle C
whose equation is x2 + y2 = 25, at A and B. Then
(a)
P is the pole of the line l with respect to
the circle C
(b)
l is the polar of the point P with respect
to the circle C
(c)
PA, PQ, PB are in AP
(d)
PQ is the HM of PA and PB
If the area of the quadrilateral formed by the
tangent from the origin to the circle
x2 + y2 + 6x – 10y +  = 0 and the pair of radii at the
points of contact of these tangents to the circle is 8
square units, then the value of  must be
(a)
2
(b)
4
(c)
16
(d)
32
C1 : x2 + y2 = 25, C2 : x2 + y2 – 2x – 4y – 7 = 0 be two
circle intersecting at A and B
(a)
Equation of common chord of C1 and C2
must be x + 2y – 9 = 0
(b)
Equation of common chord must be
x + 2y + 7 = 0
(c)
Tangents at A and B to the circle C1
1.
2.
3.
(d)
9.
10.
a2
sq. units.
6
STATEMENT-2 : In an equilateral triangle the
radius of the circle is 1/3 of the median of the
triangle.
STATEMENT-1 : If a line L = 0 is a tangent to the
circles S = 0 then it will also be a tangent to the
circle S + L = 0.
STATEMENT-2 : If a line touches a circle then
perpendicular distance from centre of the circle
on the line must be equal to the radius.
STATEMENT-1 : P is a variable point on the circle
with centre at C, CA and CB are perpendicular
from C on x-axis and y-axis respectively. The
locus of the centroid of the triangle PAB is a circle.
STATEMENT-2 : Centroid of the triangle is the
point of intersection of its medians.
STATEMENT-1 : The line x + 3y = 0 is a diameter
of thecircle x2 + y2 – 6x + 2y = 0.
STATEMENT-2 : The centre of the circle passes
through the diameter of the circle.
square inscribed in this circle is
 25 50 
intersect at  , 
 9 9 
Tangents at A and B to the circle C1
intersect at (1, 2)
The line x + y = 2 intersects the circle x2 + y2 = 3 at
two points. The equations of the straight lines
joining the origin and the points of intersection are
(a)
x – (3 + 22)y = 0
(b)
x – (3 – 22)y = 0
(c)
(3 + 22)x – y = 0
(d)
(3 – 22)x – y = 0
Both the equations x2 + y 2 + 2 x + 4 = 0 and
x2 + y2 – 4 y + 8 = 0 represent real circles if
(a)
>2
(b)
 > 2 or  < –2
(c)
  [–2, 2]
(d)
R
Assertion-Reason Type
Each question contains STATEMENT-1 (Assertion)
and STATEMENT-2 (Reason). Each question has
4 choices (A), (B), (C) and (D) out of which ONLY
ONE is correct.
(A)
Statement-1 is True, Statement-2 is True;
Statement-2 is a correct explanation
for Statement-1
(B)
Statement-1 is True, Statement-2 is True;
Statement-2 is NOT a correct
explanation for Statement-1
(C)
Statement-1 is True, Statement-2 is False
(D)
Statement-1 is False, Statement-2 is True
STATEMENT-1 : Circles are drawn having the
sides of triangle as their diameter. Radical centre
of these circles is the orthocentre of the triangle.
STATEMENT-2 : Radical axes of the circles are
the altitudes of the triangle.
STATEMENT-1 : A circle is inscribed in a
equilateral triangle of side a. The area of any
4.
5.
(Answers) EXCERCISE BASED ON NEW PATTERN
COMPREHENSION TYPE
1.
a
2.
d
3.
c
4.
d
7.
a
8.
c
9.
c
10.
a
13.
c
14.
a
MATRIX-MATCH TYPE
1.
[A-Q; B-S; C-P; D-R]
2.
[A-S; B-R; C-Q; D-P]
MULTIPLE CORRECT CHOICE TYPE
1.
a, b, c
2.
b, d
3.
b, c
4.
a
7.
a, d
8.
a, c
9.
a, b, c, d
10.
ASSERTION-REASON TYPE
1.
A
2.
A
3.
B
4.
A
Einstein Classes,
5.
11.
a
b
6.
12.
d
a
5.
a, b
a, c
6.
a, b, d
5.
A
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 18
INITIAL STEP EXERCISE
(SUBJECTIVE)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Find the equation of the circle passing through the
points (1, 2) and (3, 4) and touching the line
3x + y – 3 = 0.
Find the points on the circle x2 + y 2 = 4 whose
distance from the line 4x + 3y = 12 is 4/5 units.
Find the equation of two tangents drawn to the
circle x2 + y2 – 2x + 4y = 0 from the point (0, 1).
The circle x2 + y2 – 4x – 4y + 4 = 0 is inscribed in a
variable triangle OAB. Sides OA and OB lie along
the x and y-axis respectively, where ‘O’ is the
origin. Find the locus of the mid-point of the side
AB.
Tangents are drawn from a pt. P to the circles
x2 + y2 = a2 & x2 + y2 = b2 each. If the tangents are
perpendicular then find the locus of P.
Find the equation of the locus of the middle point
of a chord of the circle x2 + y2 = 2(x + y) such that
the pair of lines joining the origin to the point of
intersection of the chord and the circle are equally
inclined to the x-axis.
A variable circle passes through the point A(a, b)
and touches the x-axis. Show that the locus of the
other end of the diameter through A is
(x – a)2 = 4by.
Show that the locus of the point the tangents from
which to the circle x2 + y2 = a2 include a constant
angle  is (x2 + y2 – 2a2)2 tan2 = 4a2(x2 + y2 – a2).
Find the equation of the circle through the points
of intersection of the circles
x2 + y2 – 4x – 6y – 12 = 0 and
x2 + y2 + 6x + 4y – 12 = 0 and intersecting the circle
x2 + y2 – 2x – 4 = 0 orthogonally.
Two circles each of radius 5 units touch each other
at (1, 2). If the equation of their common tangent is
4x + 3y = 10, find the equation of the two circles.
Find the equation of the tangents to the circle
x2 + y2 = 4 which make an angle of 600 with the
x-axis.
Find the radius of the smaller circle which touches
3x – y = 6 at (1, –3) & also touches y = x.
The sum of squares of the tangents from a point P
to two circles is constant. Prove that the locus of P
is another circle whose centre lies midway between
the two given circles.
Obtain the equation of the circle which touches the
axis of x at (2, 0) and passes through (3, –1). Find
the equation of the other tangent through the
origin.
15.
16.
From a point on 4x – 3y = 6, tangents drawn to the
circle x2 + y2 – 6x – 4y + 4 = 0 which make an angle
 1  24 
of tan  7  between them. Find the coordinates
 
of all such points and the equations of tangents.
The base AB of a triangle is fixed and its vertex C
moves such that sin A = k sin B (k  1). Show that
the locus of C is a circle whose centre lies on the
line AB and whose radius is equal to
17.
18.
19.
20.
21.
22.
ak
(1  k 2 )
,a
being the length of the base AB.
(a)
Find the equation of the circle which
touches the x-axis at the point (3, 0) &
cuts off a chord of length 8 units on the
y-axis.
(b)
Find the equation of the circle which
touches the x-axis and passes through the
points (1, 2) and (3, 2).
Find the length of the common chord of the two
circles x 2 + y 2 – 10x – 10y = 0 and
x2 + y2 + 6x + 2y – 40 = 0.
Prove that the two circles which pass through the
points (0, a) & (0, –a) and touch the line y = mx + c
will cut orthogonally, iff c2 = a2(2 + m2)
Find the equation of all the common tangents to
the circles x2 + y2 = 25 & (x – 12)2 + y2 = 9.
Find the equation to the circles which pass through
the origin and cut off equal chords of length a from
the straight lines y = x and y = –x.
A tangent drawn at P(3, 4) on x2 + y2 – 25 = 0
intersects a variable circle of radius 5 units at A
1
1
2

 . Show that the
PA PB 5
locus of the centres of the circle is another circle.
Find the equation of that circle.
Prove that the equation of the straight line meeting
the circle x2 + y2 = a2 in two points at equal distance
d from (x 1 , y 1 ) on the curve is
and B such that
23.
1 2
d  0 . Find the equation of the
2
tangent at (x1, y1).
Two parallel tangents to a given circle are cut by a
third tangent at the points A and B. If C be the
centre of the given circle, find the ACB is a right
angle.
P(2, 2) is a point on a circle x2 + y2 – 2x – 2y = 0. If P
travels on the circle and reaches Q such that arc
xx1  yy 1  a 2 
24.
25.
1
 circumference of the cirlce then find the
6
coordinates of Q.
PQ 
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 19
FINAL STEP EXERCISE
(SUBJECTIVE)
1.
2.
3.
4.
5.
Find the equation of circle having the lines
x2 + 2xy + 3x + 6y = 0 as its normals and having size
just sufficient to contain the circle
x(x – 4) + y (y – 3) = 0.
A circle is given by 2x(x – a) + y(2y – b) = 0 (a  0,
b  0). Find the condition on a and b if two chords
each bisected by the x-axis, can be drawn to the
8.
9.
10.
11.
i
1
m
 3
( mi ) = 2
 1 ; then find the equation of the
i
circle.
13.
A circle of constant radius ‘r’ passes through the
origin ‘O’ and cuts the axes at A and B. Find the
locus of the foot of the perpendicular from O to AB
14.
The circle x2 + y2 = a2 cuts off an intercept on the
straight line lx + my = 1 which subtends an angle of
450 at the origin. Show that
15.
The line y = x touches a circle at P so that OP = 42,
where O is the origin. The point (–10, 2) lies inside
the circle and the length of the chord x + y = 0 is
62. Find the equation of the circle.
16.
From any point on the circle x2 + y2 = a2 tangents
are drawn to the circle (x – c)2 + y2 = b2 where a, b,
c are constants. Show that the locus of the middle
point of the chord of contact is a third circle.
17.
If the equation of the circles whose radii are r & R
be respectively S = 0 and S’ = 0, then prove that the
Find the intervals of values of a for which the line
y + x = 0 bisects two chords drawn from a point
Consider a family of circles passing through two
points (3, 7) and (6, 5). Show that the chords in
which the circle x2 + y2 – 4x – 6y – 3 = 0 cuts the
members of the family are concurrent. Find the
coordinates of the point.
Straight lines 5x + 12y – 10 = 0 and
5x – 12y – 40 = 0 touch a circle C1 of diameter 6
units. If the centre of C1 lies in the first quadrant,
find the equation of the circle C 2 which is
concentric with C1 and cuts intercepts of length 8
units on these lines.
The circle x2 + y2 – 4x – 4y + 4 = 0 is inscribed in a
triangle which has two of its sides along the
coordinate axes. The locus of the circumcentre of
the triangle is x + y – xy + k(x2 + y2)½ = 0. Find k.
Tangents OP and OQ are drawn from the origin O
to the circle x2 + y2 + 2gx + 2fy + c = 0. Show that
the equation of the circumcircle of the triangle OPQ
is x2 + y2 + gx + fy = 0.
Let S  x2 + y2 + 2gx + 2fy + c = 0 be a given circle.
Find the locus of the foot of the perpendicular
drawn from origin upon any chord of S which
subtends a right angle at the origin.
Einstein Classes,

 i = 1, 2, 3, 4 are four distinct points

4[a2(l2 + m2) – 1] = [a2(l2 + m2) – 2]2
 1  2a 1  2a 

 to the circle
,


2
2


2x2 + 2y2 – (1 + 2a)x – (1 – 2a)y = 0
7.
m
Find the equation of a circle through (a, 0) and
(0, b) having the smallest radius.
One of the diameters of a circle circumscribing a
rectangle ABCD is 4y = x + 7. If A and B are the
points (–3, 4) and (5, 4) respectively, find the area
of the rectangle.

1
If  m i ,
1

mi

on a circle, and if each of the distinct points on a
circle, and if each of the following conditions hold
true,
From a point P, tangents drawn to the circles
x2 + y2 + x – 3 = 0 and 4x2 + 4y2 + 8x + 7y + 9 = 0 are
equal in length. Find the equation of the circle
through P which touches the line x + y = 5 at the
point (6, –1).
 b
circle from  a,  .
 2
6.
12.
circles
18.
S S
  0 will intersect at right angles.
r R
Find the equation of the circle circumscribing the
triangle formed by the lines
x + 2y = 5 ; 2x + y = 4 & x + y = 6.
19.
If 4l 2 – 5m2 + 6l + 1 = 0, prove that the line
lx + my + 1 = 0 touches fixed circle. Find its
equation.
20.
If two curves whose equations
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 &
are
ax 2  2hxy  by 2  2g x  2f y  c  0 intersect
in four concyclic points, prove that
a  b a   b

.
h
h
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MC – 20
ANSWERS SUBJECTIVE (FINAL STEP
EXERCISE)
ANSWERS SUBJECTIVE (INITIAL
STEP EXERCISE)
1.
x2 + y2 – 3x – 7y + 12 = 0
1.
x2 + y2 + 6x – 3y = 45
4.
x + y – xy + x 2  y 2 = 0
3.
x2 + y2 – ax – by = 0
6.
x+y=2
5.
a2 > 2b2
9.
x2 + y2 + 6x + 2y – 15 = 0
6.
a  (–, –2)  (2, )
10.
x2 + y2 – 10x – 10y + 25 = 0
7.
(2, 23/3)
15.
x2 + y2 + 18x – 2y + 32 = 0
19.
x2 + y2 – 6x + 4 = 0
11.
r = 1.49
15.
7x – 24y + 102 = 0, x – 6 = 0
21.
x2 + y2 ± 2ax = 0 and x2 + y2 ± 2ay = 0
23.
xx1 + yy1 – a2 = 0
24.
/2
25.
3

  3 ,3 3
2 2 2 2 


Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111