1 objectives : 4.1 circles (a) determine the equation of a circle. (b) determine the centre and...
TRANSCRIPT
1
OBJECTIVES :
4.1 CIRCLES
(a) Determine the equation of a circle.(b) Determine the centre and radius of a
circle by completing the square(c) Find the points of intersection of two
circles(d) Find the equation of tangents and
normal to a circle(e) Find the length of a tangent from a point to a circle
4.0 CONIC SECTIONS
2
A circle is a set of all points in a plane equidistant from a given fixed pointcalled the center.
The distance from the center to any
point on the circle is called a radius.
Definition
3
The Equation of a Circle in Standard Form
1) Center (0,0) , radius : r
Equation of a circle : 222 ryx
(0,0)
P(x , y)r
Proof: OP
2OP x
y
22 00 yx
2r 2 2x y
2 2x y
If a center of the circle is not located at
the origin but at any arbitrary point
( h,k ), the equation becomes :
2 2 2x h y k r
x
y
(h,k)
4
5
2) Center ( h , k ) , radius ; r
Equation of a circle :
C (h , k)
r
y
x
Proof:
22 kyhx
P (x,y) r CP
222 rkyhx
222 kyhxr
From the Standard Equation,
2 2 2 2 22 2 0x y hx ky h k r
2 2 2( ) ( ) ..........(1)x h y k r
Expand and rearrange equation (1),
Then substitute ;2 2 2, ,h g k f c h k r
We get General Equation, 2 2 2 2 0x y gx fy c
The Equation of a Circle in General Form
6
From the General Equation2 2 2 2 0x y gx fy c
By completing the square;2 2 2 2( ) ( ) 0x g y f g f c 2 2( ) ( )x g y f 2 2g f c
Centre , ( , )C g f
Radius , 2 2r g f c
By comparing to Standard Equation
7
2 2 2( ) ( )x h y k r
( , )h k
8
Find the general equation of the circle with centre (-2,3) and radius 5 .
222 rkyhx
Example 1Example 1
SolutionSolution
Determine the center and radius of a circle
by completing the
square.
0446 22 yyxx
Example 2Example 2
SolutionSolution
; (3, 2) 3ans c r
Find the center of the circle and the length of it’s radius
2 22 2 28 12 114 0x y x y
10
Example 3 Example 3
SolutionSolution
2 2 14 6 57 0x y x y
2 22 2 28 12 114 0x y x y 2
02222 cfygxyx
compare with general equation :
; : g, f (7, 3) , 1ans centre r
Find the general equation of the circle with center (0,7) and touches the line 3x = 32 + 4y
Example 4 Example 4
SolutionSolution
C (0 ,7)
r
3 4 32 0x y
d r
2 2
ax by cd
a b
Shortest distance = d
2 2; 14 95 0ans x y y
Given that a circle passes through (9, -7) ,
(-3, -1) and (6,2 ). Find its equation.
13
Example 5 Example 5
SolutionSolution
General equation of circle2 2 2 2 0x xy yg f c
2 2; 6 8 20 0ans x y x y
Find the equation of circle passing through the points (1,1), (3,2) and with the equation of diameter y-3x+7 = 0.
15
Example 6 Example 6
SolutionSolution
(1,1)
(3,2)
General equation :2 2 2 2 0x y gx fy c
(1)At (1,1) : 2 2 2 0g f c
(2)At (3,2) : 13 6 4 0g f c
y–3x+7=0
C(-g,-f)
(1,1)
(3,2)
The diameter must passesthrough the centre,(-g ,-f )
(3)3 7 0f g
2 2; 5 4 0ans x y x y
1717
The points A and B have coordinates ( x1,
y1) and (x2 , y2). Show that the equation
of a circle where AB is the diameter of a circle is 1 2 1 2 0x x x x y y y y
Hence , find the equation of the circle
where AB is a diameter and the points of
A and B are ( 1 , 1 ) and ( 2 , 3) .
Example 7 Example 7
18
A (x1,y1) B (x2, y2)
P (x, y)
AP and PB is perpendicular :
AP PBm m 1
APm PBm 1
1
,y y
x x
2
2
y y
x x
SolutionSolution
19
A (x1,y1) B (x2,y2)
P (x,y)
1AP PBm m
2 2equation of circle; 4 3 5 0x y y x
21
Important Notes
Given the general equation of two circles: If 2( ) 4 0a b ac Two circles
intersect at two distinct points
22
2( ) 4 0b b ac Two circles touch to each other
2( ) 4 0c b ac Two circles do not intersect to each other
Find the intersection points between the two circles below:
2 2 2 24 and 4 3 0x y x y x
23
Example 8 Example 8
....... (1)
...... (2)
2 2 4x y
2 2 4 3 0x y x
SolutionSolution
7 15 7 15; , and ,
4 4 4 4ans
2 2Show that the circles 2 4 4 andx y x y 2 2 8 10 32 0 intersect at two
distinct points.
Hence, find the intersection points.
x y x y
25
Example 9 Example 9
2 2 2 4 4 x y x y 2 2 8 10 32 0x y x y
....... (1)
....... (2)
26
SolutionSolution
; (1,5) and (4,2)ans
28
(1) Standard equation : x2 + y2 = r2
The equation of tangents to a circle at
the point P ( x1, y1) is given by
xx1 + yy1 = r2
0
P(x1 , y1)
x
y
The equations of tangents and normal to a circle
xx1 + yy1 = r2
29
(2) General equation :
xx1 + yy1 + g (x + x1) + f (y + y1) +c = 0
2 2 2 2 0x y gx fy c
The equation of tangents to a circle at
the point P ( x1, y1) is given by
Find tangent and normal line to the circlex2 + y2 = 5 at the point ( -2, 1).
30
Example 10 Example 10
SolutionSolution
The equation of tangent at ( -2,1)
xx1 + yy1 = r2
-2x + y(1) = 5
y = 2x + 5
31
31
2tm
The equation of normal at ( -2,1)
1y
nm
2x 1
2
1
2
1 1( )ny y m x x
2 2 ( 2)y x
2 0x y
32
Find the equation of the tangent and normal of the circle at the point (3,1)
2 2 8 6 8 0x y x y
Example 11 Example 11
SolutionSolution
2 2 2 2 0x y gx fy c
2 2 8 6 8 0x y x y
Equation of tangent at (3,1),
; 4 1 0ans y x
Equation of normal at (3,1),
; 4 7ans y x
34
TheoremThe length of the tangent from a fixed point P(x1, y1) to a circle with equation
x2 + y2 +2gx + 2fy+ c = 0 is given by
C
r
P (x1,y1)
d
Q
The length of a tangent from a point to a circle
2 21 1 1 12 2PQ d x y g x f y c
Find the length of the tangent from the point A (6,7) to a circle x2 + y2 – 2x – 8y = 8
Example 12 Example 12
SolutionSolution
; 3ans