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Transcript
Co-ordinate Geometry of the Circle
Notes
Aidan Roche
2009
(c) Aidan Roche 2009
1
Given the centre and radius of a circle, to find
the equation of Circle K?
K
r
c(h, k)
Method
• Sub centre & radius into:
(x – h)2 + (y – k)2 = r2
• If required expand to:
x2 + y2 +2gx +2fy + c = 0
(c) Aidan Roche 2009
2
To find the centre and radius. Given the Circle
K: (x – h)2 + (y – k)2 = r2
K
r
c
Method
• Centre: c(h, k)
• Radius = r
(c) Aidan Roche 2009
3
To find the centre and radius. Given the Circle
K: x2 + y 2 = r2
K
r
c
Method
• Centre: c(0, 0)
• Radius = r
(c) Aidan Roche 2009
4
To find centre and radius of K. Given the circle
K: x2 + y2 +2gx +2fy + c = 0?
K
r
c
Method
• Centre: c(-g, -f)
2
2
r

g

f
c
• Radius:
(c) Aidan Roche 2009
5
Given equation of circle K, asked if a given
point is on, inside or outside the circle?
a
b
c
K
Method
• Sub each point into the
circle formula K = 0
Answer > 0
Answer = 0
Answer < 0
(c) Aidan Roche 2009
outside
on
inside
6
Important to remember
d
Theorem
• Angle at centre is
twice the angle on
the circle standing
the same arc
θ
c
2θ
a
b
(c) Aidan Roche 2009
7
Important to remember
90o
diameter
Theorem
• Angle on circle
standing the
diameter is 90o
(c) Aidan Roche 2009
8
To find equation of circle K given end points
of diameter?
K
a
c
r
b
Method
• Centre is midpoint [ab]
• Radius is ½|ab|
• Sub into circle formula
(c) Aidan Roche 2009
9
To prove a locus is a circle?
Method
•
r
c
K
•
If the locus of a set of
points is a circle it can be
written in the form:
x2 + y2 +2gx + 2fy + c = 0
We then can write its
centre and radius.
(c) Aidan Roche 2009
10
To find the Cartesian equation of a circle
given Trigonometric Parametric equations?
Method
•
Trigonometric equations
of a circle are always in
the form:
x = h ± rcosѲ
y = k ± rsinѲ
•
Sub h, k and r into
Cartesian equation:
(x – h)2 + (y – k)2 = r2
r
c
K
(c) Aidan Roche 2009
11
To prove that given Trigonometric Parametric
equations (x = h ± rcosѲ, y = k ± rsinѲ)
represent a circle?
Method
r
c
K
•
•
•
•
Rewrite cosѲ (in terms of x, h & r)
and then evaluate cos2Ѳ.
Rewrite sinѲ (in terms of y, h & r)
and then evaluate sin2Ѳ.
Sub into: sin2Ѳ + cos2Ѳ = 1
If it’s a circle this can be written
in the form:
x2 + y2 +2gx + 2fy + c = 0
(c) Aidan Roche 2009
12
To find the Cartesian equation of circle
(in the form: x2 + y2 = k)
given algebraic parametric equations?
Method
r
c
• Evaluate: x2 + y2
• The answer = r2
• Centre = (0,0) & radius = r
K
(c) Aidan Roche 2009
13
Given equations of Circle K and Circle C, to
show that they touch internally?
C
r1
K
c1
r2
c2
d
Method
• Find distance
between centres
• If d = r1 - r2 QED
(c) Aidan Roche 2009
14
Given equations of Circle K and Circle C, to
show that they touch externally?
C
K
r1
r2
d
c2
Method
• Find distance d
between centres
• If d = r1 + r2 QED
c1
(c) Aidan Roche 2009
15
Given circle K and the line L to find points of
intersection?
b
L
Method
• Solve simultaneous equations
a
K
(c) Aidan Roche 2009
16
Important to remember
K
90o
t
radius
c
Theorem
• A line from the
centre (c) to the
point of tangency (t)
is perpendicular to
the tangent.
Tangent
(c) Aidan Roche 2009
17
Important to remember
b
d
a
radius
90o
c
Theorem
• A line from the
centre perpendicular
to a chord bisects
the chord.
(c) Aidan Roche 2009
18
Given equation of Circle K and equation of
Tangent T, find the point of intersection?
T
K
t
Method
• Solve the simultaneous
equations
(c) Aidan Roche 2009
19
Given equation of Circle K and asked to find
equation of tangent T at given point t?
T
Method 1
• Find slope [ct]
• Find perpendicular slope of T
• Solve equation of the line
t
c
Method 2
K
• Use formula in log tables
(c) Aidan Roche 2009
20
To find equation of circle K, given that x-axis
is tangent to K, and centre c(-f, -g) ?
Method
K
c(-g, -f)
• On x-axis, y = 0 so t is (-f, 0)
• r = |f|
• Sub into circle formula
r = |f|
X-axis
t(-g, 0)
(c) Aidan Roche 2009
21
To find equation of circle K, given that y-axis
is tangent to K, and centre c(-f, -g) ?
r = |g|
t(0, -f)
y-axis
c(-g, -f)
Method
• On y-axis, x = 0 so t is (0, -g)
• r = |g|
• Sub into circle formula
K
(c) Aidan Roche 2009
22
Given equation of Circle K and equation of
line L, how do you prove that L is a tangent?
Method 1
L
K
r
• Solve simultaneous
equations and find that
there is only one solution
Method 2
c
• Find distance from c to L
a (  g )  b(  f )  c
d
a 2  b2
• If d = r it is a tangent
(c) Aidan Roche 2009
23
Given equation of Circle K & Line L: ax + by + c = 0
to find equation of tangents parallel to L?
L
Method 1
T1
r
c
K
• Find centre c and radius r
• Let parallel tangents be:
ax + by + k = 0
• Sub into distance from point
to line formula and solve:
a (  g )  b(  f )  k
r
a 2  b2
r
T2
(c) Aidan Roche 2009
24
Given equation of Circle K and point p, to find
distance d from a to point of tangency?
Method
T
• Find r
• Find |cp|
• Use Pythagoras to find d
t
d?
r
c
|cp|
p
K
(c) Aidan Roche 2009
25
Given equation of Circle K and point p, to find
equations of tangents from p(x1,y1)?
T1
Method 1
r
K
c
r
T2
p
• Find centre c and radius r
• Sub p into line formula and write
in form T=0 giving:
mx – y + (mx1 – y1) = 0
• Use distance from point to line
formula and solve for m:
m (  f )  1(  g )  ( mx1  y1 )
r
m 2  12
(c) Aidan Roche 2009
26
Given equation of Circle K and Circle C, to
find the common Tangent T?
T
C
Method
• Equation of T is:
K–C=0
K
(c) Aidan Roche 2009
27
Given equation of Circle K and Circle C, to
find the common chord L?
L
Method
• Equation of T is:
K–C=0
K
C
(c) Aidan Roche 2009
28
Given three points and asked to find the
equation of the circle containing them?
b
a
Method
• Sub each point into formula:
x2 + y2 + 2gx + 2fy + c = 0
• Solve the 3 equations to find:
g, f and c,
• Sub into circle formula
c
(c) Aidan Roche 2009
29
Given 2 points on circle and the line L
containing the centre, to find the equation of
the circle?
b
a
Method
• Sub each point into the circle:
x2 + y2 + 2gx + 2fy + c = 0
• Sub (-g, -f) into equation of L
• Solve 3 equations to find: g, f and c,
• Sub solutions into circle equation
L
(c) Aidan Roche 2009
30
Given the equation of a tangent, the point of
tangency and one other point on the circle,
to find the equation of the circle?
L
a
Method
b
T
• Sub each point into the circle:
x2 + y2 + 2gx + 2fy + c = 0
• Use the tangent & tangent point to
find the line L containing the centre.
• Sub (-g, -f) into equation of L
• Solve 3 equations to find: g, f and c,
• Sub solutions into circle equation
(c) Aidan Roche 2009
31