Circle PowerPoint

1
Chapter 3 Conics
3.3
The Circle
3.3.1
MATHPOWERTM 12, WESTERN EDITION
2
Developing the Standard Forms of the Equation of
a Circle
Note OP is the radius of the circle.
P(x, y)
O(0, 0)
The standard form of the equation of a circle
with its centre at the origin (0, 0) is x2 y2
r2.
3.3.2
3
Developing the Standard Forms of the Equation of
a Circle
P(x, y)
C(h, k)
This is the standard form of the equation of a
circle with the centre at (h, k).
3.3.3
4
Finding the Equation of a Circle
Determine the equation of a circle with centre
C(-5, 2) and passing through the point P(-8, 7).
(h, k)
(x, y)
From the general form (x - h)2 (y - k)2 r2
Substitute the values of h and k from C(-5, 2)
(x - (-5))2 (y - 2)2 r2 (x 5)2 (y -
2)2 r2
(x 5)2 (y - 2)2 r2 (-8 5)2 (7 - 2)2
r2 9 25 r2
34 r2
Use the point P(-8, 7) to find the value of r2
Therefore, the equation of the circle in standard
form is (x 5)2 ( y - 2)2 34.
3.3.4
5
Writing the General Form of the Equation of a
Circle
The general form of the equation is
Ax2 Cy2 Dx Ey F 0.
Write the following equation in general form
(x 5)2 (y - 2)2 34
(x 5)2 (y - 2)2 34 x2 10x
25 y2 - 4y 4 34 x2 y2 10x - 4y
29 34 x2 y2 10x - 4y - 5 0
3.3.5
6
Finding the Centre and the Radius
Find the centre and the radius of each circle
1. x2 y2 - 8x 10y - 14 0
To find the centre and radius, write the equation
in standard form. To do this, you must complete
the square
x2 y2 - 8x 10y -
14 0 (x2 - 8x _____ ) (y2 10y _____)
14 _____ _____
16
16
25
25
(x - 4)2 (y 5)2 55
The centre is (4, -5) and the radius is 7.4.
2. 3x2 3y2 6x 12y 5 0
(3x2 6x) (3y2
12y) -5 3(x2 2x _____) 3(y2 4y _____)
-5 _____ _____
1
3
12
4
3(x 1)2 3(y 2)2 10
The centre is (-1, -2) and the radius is
3.3.6
7
Using a Graphing Calculator
Graph (x - 3)2 (y - 4)2 16
Your calculator will only graph a function,
therefore, you must write the equation in the
form y .
Make sure that you use a ZSquare graphing
window. You can also use the Draw circle command
on your TI-83 Press 2ndPRGM 9 and enter the
following Circle (3, 4, 4)
3.3.7
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Using a Graphing Calculator
Using your graphing calculator, graph the
following equations
a) x2 y2 16 b) 4x2 y2 16 c) 0.5x2
y2 16 d) Ax2 y2 16, when A 0
x2 y2 16
Ax2 y2 16, when A 0
x2 y2 16
x2 y2 16
x2 y2 16
4x2 y2 16
0.5x2 y2 16
4x2 y2 16
0.5x2 y2 16
4x2 y2 16
3.3.8
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Using a Graphing Calculator contd
Using your graphing calculator, graph the
following equations
d) x2 4y2 16 e) x2 0.5y2 16 f) x2 Cy2
16, when C 0
x2 Cy2 16, when C 0
x2 y2 16
x2 y2 16
x2 y2 16
x2 4y2 16
x2 4y2 16
x2 4y2 16
x2 0.5y2 16
x2 0.5y2 16
3.3.9
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Using a Graphing Calculator contd
Conclusions
The values of A and C affect the graph of the
circle by either a vertical or horizontal
compression or expansion
A gt 1 results in a horizontal compression. 0 lt A
lt 1 results in a horizontal expansion. A 0
results in a pair of horizontal parallel lines.
C gt 1 results in a vertical compression. 0 lt C lt
1 results in a vertical expansion. C 0 results
in a pair of vertical parallel lines.
3.3.10
11
Assignment
Suggested Questions
Pages 141 and 142 A 1-25 odd, 27-32 B
36-45, 49, 50, 51, 54, 56, 58 (graph),
62
3.3.11