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PARABOLA
1.
Let S be a given fixed point (focus) and let l be given fixed line (Directrix). Let SP and
PM be the distance of a variable point P to the focus and directrix respectively and P
moves such that
SP
= e (constant > 0) (eccentricity) then locus of P is called a conic
PM
or conic section.
2.
If e = 1, the conic is called a parabola.
If e < 1, the conic is called an ellipse.
If e > 1, the conic is called a hyperbola.
3.
The general equation of a conic is S = ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 (Second
degree equation in x and y).
i) If
0, h2 = ab, then S = 0 represents a parabola.
ii) If
0, h2<ab, then S = 0 represents an ellipse.
iii) If
0, h2>ab, then S = 0 represents a hyperbola.
iv) If
4.
0, h2>ab, a + b = 0, then S = 0 represents a rectangular hyperbola.
standard forms of the parabola
S. No. Content
Equation
I
II
y2 = 4ax
Y
Figure
X
Z
A
y2 = –4ax
L
L
X
IV
x2 = 4ay
L
S
Y
III
x2 = –4ay
L
X
A
S
L
Y
Y
Y
Z
X
Y
X
S
A
Z
Y
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Z
L
X
X
L
A
X
S
Y
L
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1.
Vertex (A)
(0, 0)
2.
Focus (S)
3.
Point of intersectionof axis and directrix (Z)
(a, 0)
(–a, 0)
4.
(0, 0)
(0, 0)
(0, 0)
(–a, 0)
(0, a)
(0, –a)
(0, –a)
(0, a)
(a, 0)
End points of latus rectum (L, L )
(a,
2a)
(–a,
5.
Equation of axis
y=0
6.
Equation of Directrix x = –a
7.
Equation of tangent at vertex
x=a
x=0
x=0
y = –a
y=a
y=0
x = –a
Length of latusrecturm (LL )
4a
10.
Distance from focus to directrix (SZ)
11.
SA = AZ
5.
x=0
y=0
Equation of latus rectum
x=a
9.
( 2a, –a)
( 2a, a)
y=0
x=0
8.
2a)
a
y=a
y = –a
4a4a4a
2a
2a2a2a
aaa
i) If the axis of a parabola is parallel to x-axis, equation of the parabola will be of the
form (y – )2 = 4a(x – ) (or)
(y – )2 = –4a(x – ) (or) x = ay2 + by + c.
ii) If the axis of the parabola is parallel to y-axis, equation of the parabola will be of
the form (x – )2 = 4a(y – ) (or) (x – )2 = –4a(y – ) (or) y = ax2 + bx + c.
6.
In the equation of the parabola (y – )2 = 4a(x – ).
i) Vertex = ( , ).
ii) Focus = ( + a, )
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iii) Ends of latustrectum = ( + a,
2a)
iv) Equation of axis is y =
– a.
v) Equation of directrix is x =
vi)Equation of latustrectum is x =
+a
vii) Length of latustrectum = 4a.
7.
In the equation of the parabola (x – )2 = 4a(y – )
i) Vertex = ( , )
ii) Focus = ( ,
+ a)
iii) Ends of latustrectum = (
2a,
+ a)
iv)Equation of axis is x =
v) Equation of directrix is y =
–a
vi)Equation of latustrectum is y =
+a
vii) Length of latustrectum = 4a.
8.
The focal distance of the point P(x1, y1) on the parabola
i) y2 = 4ax is SP = |x1 + a|
ii) x2 = 4ay is SP = |y1 + a|.
9.
A chord of the parabola perpendicular to its axis is called double or dinate of the
parabola.
10.
A chord of the parabola which is passing through focus is called focal chord.
11.
The focal chord of the parabola which is perpendicular to axis is called latus rectum.
12.
Equation of Tangent to y2 = 4ax at (x1, y1) is S1 = 0.
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13.
The condition that the line y = mx + c may be a Tangent to the parabola y2 = 4ax is
c=
14.
a
and the point of contact is a2 , 2a .
m
m m
The condition for the line lx + my + n = 0 to be a Tangent to parabola y2 = 4ax is
am2 = ln and the point of contact is
15.
n 2am
,
l l
The condition that the line lx + my + n = 0 may be a Tangent to the parabola x 2 = 4ay
2al n
,
m m
is l2 = mn and the point of contact is
16.
.
.
The equation of Tangent to y2 = 4ax in slope from is y = mx +
a
m
(or)
m2x – my + a = 0.
17.
If m1 and m2 are the slopes of tangents from an external point (x1, y1) to the Parabola
y2 = 4ax then they are the roots of m2x1 – my1 + a = 0 and hence m1 + m2 =
m1m2=
18.
If
a
.
x1
is the acute angle between Tangents drawn from (x 1, y1) to parabola
S = y2 – 4ax = 0 then Tan =
19.
y1
and
x1
S11
| x1
a|
.
Locus of the point of the intersection of perpendicular tangents drawn to the parabola
is its directrix.
20.
The angle between the tangents drawn from a point on the directrix to be parabola is
90o.
21.
Tangents drawn at the ends of focal chord of a parabola are at right angles, they
intersect on directrix.
22.
The angle between the tangents drawn at the ends of latusrectum of the parabola
y2 = 4ax is 90o and the point of intersection of these tangents is (–a, 0).
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23.
The angle between the normals drawn at the ends of latusrectum of the parabola
y2 = 4ax is 90o and the point of intersection of these normals is (3a, 0).
24.
The Tangents and normals at the ends of latusrectum of parabola y2 = 4ax forms a
square whose side is 2 2 a and area is 8a.
a
m
25.
For the parabola y2 = 4a(x + a), equation of tangent in slope from is y = m(x + a) +
26.
Equation of common tangent to two parabolas y2 = 4ax and x2 = 4by is
a1/3x + b1/3y + (ab)2/3 = 0.
27.
Equation of chord of contact of (x1, y1) to y2 = 4ax is S1 = 0.
28.
Pole of line lx + my + n = 0 w,r.t. to parabola y2 = 4ax is.
n 2am
,
l
l
29.
Pole of line lx + my + n = 0 w.r.t. to parabola x2 = 4ay is
2al n
,
m m
30.
The condition for two points (x1, y1) and (x2, y2) to be conjugate to parabola y2 = 4ax
.
.
is S12 = 0.
31.
The condition that the lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 to be conjugate
w.r.t. to y2 = 4ax is l1n2 + l2n1 = 2am1m2.
32.
The condition that the lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 to be conjugate
w.r.t. to x2 = 4ay is m1n2 + m2n1 = 2al1l2.
33.
If (x1, y1) is midpoint of chord of y2 = 4ax then equation of chord is S1 = S11.
34.
For any curve y = f(x), the slope of chord having (x1, y1) as middle point is
35.
dy
dx
.
at x1 , y1
Equation of pair of tangents drawn from an external point (x 1, y1) to parabola y2 = 4ax
is S12 = S.S11.
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Parametric Form:
36.
For all values of t, the point (at2, 2at) lies on parabola y2 = 4ax and it is denoted by ‘t’.
Equations x = at2, y = 2at are called parametric equation of y2 = 4ax.
37.
The focal distance of a point P(at2, 2at) on the parabola y2 = 4ax is |at2 + a|.
38.
If (at2, 2at) is one end of the double ordinate of y2 = 4ax then its length = 4at.
39.
Equation of chord joining t1 and t2 on parabola y = 4ax is y(t1 + t2) = 2x + 2at1t2.
40.
If t1, t2 are the ends of the focal chord of a y2 = 4ax, then t1t2 = –1.
41.
If (at2, 2at) is one end of focal chord of the parabola y2 = 4ax then its other end is
a
2a
,
2
t
t
.
1
t
42.
Slope of the tangent at ‘t’ on y2 = 4ax is .
43.
Equation of tangent at ‘t’ to y2 = 4ax is yt = x + at2.
44.
The point of intersection of tangents at t1 and t2 on y2 = 4ax is [at1t2, a(t1 + t2)].
45.
Slope of the normal at ‘t’ on y2 = 4ax is –t.
46.
Equation of normal at ‘t’ on y2 = 4ax is y + xt = 2at + at3. Since this is cubic equation
in t, it has 3 roots in which at least one of them is real. Therefore from a given point,
we can draw at most three normals to a parabola.
47.
If t1, t2, t3 are the feet of the normals drawn from a point (x1, y1) to the parabola
y2 = 4ax then they are the roots of at3 + (2a – x1)t – y1 = 0 and hence t1 + t2 + t3 = 0,
t1t2 + t2t3 + t3t1 =
2a - x 1
y
, t1t2t3 = 1 .
a
a
48.
The tangent at end of focal chord of parabola is parallel to normal at the other end.
49.
Equation of normal to y2 = 4ax in slope form is y = mx – 2am – am3 where ‘m’ is
slope of normal.
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50.
The condition that the line lx + my + n = 0 to be a normal to parabola y2 = 4ax is
al3 + 2alm2 + m2n = 0.
2
.
t1
51.
If the normal at ‘t1’ on y2 = 4ax meets it again at ‘t2’ then t2 = – t1–
52.
If the normals at t1 and dt2 on the parabola y2 = 4ax meet again on parabola at t3 then
t1t2 = 2 and t1 + t2 + t3 = 0.
53.
If the normal chord at ‘t’ on y2 = 4ax subtends a right angle at the vertex then t2 = 2.
54.
If the normal chord at ‘t’ on y2 = 4ax subtends a right angle at the focus then t2 = 4.
Length of Chord:
55.
If P(x1, y1), Q(x2, y2) are the ends of chord of a curve then its length PQ = |x 1 – x2|
1 m 2 where ‘m’ is slope of PQ .
56.
If t1 and t2 are ends of chord of y2 = 4ax then its length = a|t1 – t2| t1 t 2
57.
1
The length of focal chord drawn at a point ‘t’ on the parabola y = 4ax is a t
t
58.
If a focal chord of a parabola y2 = 4ax makes an angle
2
4.
2
2
.
with its axis then its length
= 4a cosec2 .
3/ 2
59.
4a 1 t 2
Length of normal chord at ‘t’ on y = 4ax is
t2
60.
Length of chord of contact of (x1, y1) w.r.t. to y2 = 4ax is
61.
Length of the chord of y2 = 4ax having (x1, y1) as its mid pointis
2
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.
1
S11 y12
a
1
a
4a 2 .
S11 y12
4a 2 .
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Area of the Triangle:
62.
Area of triangle formed by two tangents from (x1, y1) and its chord of contact is
S11 3 / 2
.
2a
63.
Area of triangle inscribed in parabola y2 = 4ax is
1
y1
8a
y2 y2
y3 y3
y1
where
y1 – y2 and y3 are ordinates of angular points.
64.
Area of the triangle formed by the tangents at three points whose the ordinates y1, y2,
y3 on y2 = 4ax is
1
y1
16a
y2 y2
y3 y3
y1 .
Extra Information:
65.
If PSQ is a focal chord of a conic then
latusrectum of a conic.
1
SP
1
SQ
2
when l is length of semi
l
(Or)
The semi latus rectum is harmonic mean between segments of focal chord of the
parabola.
66.
If SP and SQ are the distances of two point P and Q on parabola y2 = 4ax from focus
S, T is pint on intersection of tangent at P and Q then St2 = SP .SQ.
67.
Least length of focal chord of y2 = 4ax is 4a.
68.
The length of the normal chord of the parabola is least when it subtends a right angle
at the vertex.
69.
The orthocentre of the triangle formed by three tangents of a parabola lies on a
direction.
70.
The circle passing through three feet of the normals drawn from a point to the
parabola.
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71.
The circum circle of the triangle formed by any three tangents to a parabola passes
through the focus of the parabola.
72.
P is a point on the parabola whose focus is S and PN is the perpendicular drawn from
P to the directrix, then the tangent at P is the internal bisector P and normal is the
external bisector of P.
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