1. Chapter 3 Conics
3.3
The Circle
MATHPOWERTM 12, WESTERN EDITION
3.3.1

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
= 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 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:
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 25 16 25
(x - 4)2 + (y + 5)2 = 55
The centre is (4, -5) and the radius is 7.4.
x2 + 3y2 + 6x + 12y + 5 = 0
(3x2 + 6x) + (3y y) = -5
3(x2 + 2x + _____) + 3(y2 + 4y + _____) = -5 + _____ + _____ 1 4 3 12
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 [2nd][PRGM] 9 and
enter the following:
Circle (3, 4, 4)
3.3.7

8. 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

9. Using a Graphing Calculator [cont’d]
Using your graphing calculator,
graph the following equations:
d) x2 + 4y2 = 16
e) x y2 = 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
x y2 = 16
x y2 = 16
3.3.9

10. Using a Graphing Calculator [cont’d]
Conclusions:
The values of A and C affect the graph
of the circle by either a vertical or horizontal
compression or expansion:
A > 1 results in a horizontal compression.
0 < A < 1 results in a horizontal expansion.
A = 0 results in a pair of horizontal parallel lines.
C > 1 results in a vertical compression.
0 < C < 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 , 49,
50, 51, 54, 56,
58 (graph), 62
3.3.11