Circles

Circles

Equation of a circle: ( x – h )2 + ( y – k )2 = r2

Center of the circle: C( h, k )

Radius of the circle: r

Diameter of the circle: d = 2r

rr

d= 2r

Example 5: Find the center and the radius of a circle whose equation is

( x – 3)2 + ( y + 5 )2 = 36.

Solution:

Center: C( 3, -5)

Radius: r = 6

Example 6: Graph the above Circle.6

66

6 (3,-5)(9, -5)(-3, -5)

(3, -11)

(3, 1 )

x

y

Graphing Semi CirclesUpper half, Lower half, right half, and left half

Let us find the equations of the upper half, lower half, right half and left half of the circle x 2 + y2 = 25.

x2 + y2 = 25 is a circle with center ( 0, 0 ) and radius r = 5. The graph of this circle is shown below.

To find upper and lower halves, we solve for y in terms of x.

2522 yx22 25 xy

225 xy

025 2 xy1) Represents the upper half plane

025 2 xy

-5 5

5025 2 xy

5-5

-5

5

0

-5

5-5

025 2 xy

Continued…

To find right and left halves, we solve for x in terms of y.

-5

5

5 025 2 yx 025 2 yx

3) Represents the right half plane

025 2 yx 4) Represents the left half plane

025 2 yx

The End