Circles Notes

1st Day

A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius of the circle, and the given point is the center of the circle.

Standard form of a circle with center C(h, k) and radius r is

The standard form of the equation of a circle with center at the origin (0, 0) and radius r is .

Standard Equation

2 2 2 x h y k r

2 2 2 x y r

Express in standard form the equation of the circle centered at (−2, 3) with radius 5.

Example 1

2 2 2

2 2 2

2 2

2 3 5

2 3 25

x h y k r

x y

x y

Express in standard form the equation of the circle with center at the origin and radius of 4. Sketch the graph.

Example 2

2 2

2 2 2

2

2 2

4

16

x y

x y

x y r

-8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

Find the center and radius of the circle with the equation

Example 3

2 2( 5) ( 1) 16 x y

2 2

22 25 1 4

center 5, 1 ; radius 4

x h y k r

x y

Find the center and radius of the circle with the given equation

To find the center and radius you must rewrite this equation into standard form by completing the square twice.

Example 4

2 2 4 6 5 0 x y x y

2 2

2 2

22 2 2

2 2

2 2

4 6 5 0

4 ____ 6 ____ 5 ____ ____

4 2 6 3 5 4 9

4 4 6 9 8

2 3 8

center 2, 3 ; radius 8 2 2

x y x y

x x y y

x x y y

x x y y

x y

Find the center and radius of the circle with the given equation

Example 5

2 2 6 10 4 0 x y x y

2 2

2 2

22 2 2

2 2

2 2

6 10 4 0

6 ____ 10 ____ 4 ____ ____

6 3 10 5 4 9 25

6 9 10 25 38

3 5 38

center 3, 5 ; radius 38

x y x y

x x y y

x x y y

x x y y

x y

The graph of x2 + y2 = 16 is translated 3 units to the right and 2 units down. Find the equation of the new circle and sketch its graph.

Example 6

22x 3 y 2 16

-8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

2nd Day

Write the equation for each circle described.

Example 1

a. The circle has its center at (8, -9) and passes through the point at (4, -6).

2 2 28 9 x x r

22 28 4 9 6 r216 9 25 r

2 28 9 25 x x

b. The endpoints of a diameter are at (1, 8) and (1, −4).

8 41 1center = , 1, 6

2 2

2 22 1 1 8 6 0 4 4 r

2 21 6 4 x x

The circle with center (3, 4) and tangent to the y–axis.

If a circle is tangent to a line, then a radius of the circle is perpendicular to the line.

r = 3

-8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

2 23 4 9 x y

Find the coordinates of the center and the radius of

Do completing the square twice.

Example 2

2 22 12 2 4 12 x x y y

2 22 6 _ 2 2 _ 12 2 _ 2 _ x x y y

2 22 12 2 4 12 x x y y

2 22 6 9 2 2 1 12 2 9 2 1 x x y y

2 22 3 2 1 8 x y

2 23 1 4 x y

center = (3, 1); radius = 2