Honors Pre-Calculus

Name ________________________

Circles

Circle: Set of all points in a plane which are the same distance (radius) from a fixed point (center).

Standard form for the equation of a circle:

−

+

−

=

2

2

2

(

x

h

)

(

y

k

)

r

circle with center (h, k) and radius r.

General form for the equation of a circle:

+

+

+

+

=

2

2

ax

bx

cy

dy

e

0

We use the Completing the Square method to convert equations from general form to standard form.

Ex1:

2

−12x + 3y

2

+ 30y + 75 = 0

3x

2

−12x + 3y

2

+ 30y = −75

3x

(DIVIDE both sides by 3)

2

− 4 x + y

2

+ 10y = −25

x

2

− 4 x + __ ) + (y

2

+ 10y + __ ) = −25 + __ + __

(x

2

− 4 x + 4) + (y

2

+ 10y + 25) = −25 + 4 + 25

(x

(x − 2)

2

+ (y + 5)

2

= 4

This circle is centered on (2, -5) and the radius = 2

Complete the square to rewrite each equation in to standard form for a circle. Graph each circle.

+

+

−

+

=

−

+

−

−

=

2

2

2

2

1.

x

y

8

x

16

y

31

0

2.

5

x

10

x

5

y

30

y

75

0

+

−

+

=

+

−

+

+

=

2

2

2

2

3.

4

x

4

y

16

y

15

0

4.

2

x

2

y

24

x

36

y

162

0

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