1.5 – Circles (Lesson Objectives)

• Write the standard form of the equation of a circle.

• Graph a circle by hand and with a calculator using the standard form of the equation of a circle.

• Work with the general form of an equation of a circle.

DefinitionA Circle is a set of points in the xy plane that are a fixed distance r from a fixed point (h,k), where (h,k) is the center of the circle and r is the radius.

Standard Form of a Circle

Center is at (h, k) 2 2 2x h y k r

r is the radius of the circle

Every binomial squared has been multiplied out.

Every term is on the left side, set equal to 0.

2 2: 4 6 3 0 Example x y x y

General Form of a Circle

EX 1 Write the standard form for the equation of a circle with center (3, -2) and a radius of 4.

2 2 2+x h y k r

22 23 + 2 4x y

2 23 + 2 16x y

EX 2 Write an equation of a circle with center (-4, 0) and a diameter of 10.

2 2 2+x h y k r

2 2 24 + 0 5x y

2 24 +y 25x

EX 3 Write an equation of a circle with center (2, -9) and a radius of .

11

2222 9 11x y

2 2 2+x h y k r

2 22 + y+9 11x

EX 4 Find the coordinates of the center and the measure of the radius.

2 2 26 + 3 25x y

The center is

The radius is

The equation is

5. Find the center, radius, & equation of the circle.

(0, 0)

12

x2 + y2 = 144

The center is

The radius is

The equation is

6. Find the center, radius, & equation of the circle.

(1, -3)

7

(x – 1)2 + (y + 3)2 = 49

7. Graph the circle, identify the center & radius.(x – 3)2 + (y – 2)2 =

9

Center (3, 2)

Radius of 3

Converting from General Form to Standard

1.Move the x terms together and the y terms together.

2.Move C to the other side.3.Complete the square (as

needed) for x.4.Complete the square(as needed)

for y.5.Factor the left & simplify the

right.

2 2( 8 ) 7 x x y

8. Write the standard equation of the circle. State the center & radius.

2 2 8 7 0x y x

Center: (4, 0) radius: 3

2 2( 4) 9x y

2 2( 8 )16 167 x x y

2 24 6 3 x x y y

9. Write the standard equation of the circle. State the center & radius.

2 2 4 6 3 0 x y x y

Center: (-2, 3) radius: 4

2 22 3 16 x y

2 24 6 9394 4 x x y y

2 28 2 10 x x y y

10. Write the standard equation of the circle. State the center & radius.

2 22 2 16 4 20 0 x y x y

2 24 1 7 x y

2 28 2 1 116 1610 x x y y

2 2 8 2 10 0 x y x y

: (4, 1) : 7 2.6 Center Radius

11. Write the general form of the equation of the circle.

2 24 3 36 x y

2 28 16 6 9 36 x x y y2 28 16 6 9 36 0 x x y y

2 2 8 6 11 0 x y x y

Homework• 1.5 - p.49 1-4, 7-43

by 4’s, 45• Bring in textbook on

Monday• Bring in your

Notecard for Tuesday’s Ch.1 Test