10.3 - Circles

Find the missing value to complete the square.

6. x2 – 2x + 7. x2 + 4x + 8. x2 – 6x +

Circles – Warm Up

Find the missing value to complete the square.

6. x2 – 2x + 7. x2 + 4x + 8. x2 – 6x +

Simplify.

1. 16 2. 49 3. 20 4. 48 5. 72

Solutions

6. x2 – 2x + ; c = = – = (–1)2 = 1

7. x2 + 4x + ; c = = = 22 = 4

8. x2 – 6x + ; c = = – = (–3)2 = 9b2

b2

b2

22

42

62

2 2

2 2

2 2b2b2b2

b2

1. 16 = 4

2. 49 = 7

3. 20 = 4 5 = 2 5

4. 48 = 16 3 = 4 3

5. 72 = 36 2 = 6 2

CIRCLE TERMS

EQUATION FORM

CENTER

RADIUS

MIDPOINT

FORMULA

DISTANCE

FORMULA

(x – h)² + (y – k)² = r²

(h, k )

r

2

yy,

2

xx 2121

212

212 )yy()xx(

r

C=(h , k)

Definition: A circle is an infinite number of points a set distance away from a center

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

Circles

(x – h)2 + (y – k)2 = r2 Use the standard form of the equation of a circle.

(x – 3)2 + (y – (–2))2 = 32 Substitute 3 for h, –2 for k, and 3 for r.

(x – 3)2 + (y + 2)2 = 9 Simplify.

Check: Solve the equation for y and enter both functions into your graphing calculator.

(x – 3)2 + (y + 2)2 = 9

(y + 2)2 = 9 – (x – 3)2

y + 2 = ± 9 – (x – 3)2

y = –2 ± 9 – (x – 3)2

Write an equation for the translation of x2 + y2 = 16 two units

right and one unit down.

Circles

(x – 2)2 + (y – (–1))2 = 16 Substitute 2 for h, –1 for k, and 16 for r 2.

(x – h)2 + (y – k)2 = r 2 Use the standard form of the equation of a circle.

(x – 2)2 + (y + 1)2 = 16 Simplify.

The equation is (x – 2)2 + (y + 1)2 = 16.

WRITE and GRAPH

• A) write the equation of the circle in standard form

• x² + y² - 4x + 8y + 11 = 0• Group the x and y terms• x² - 4x + y² + 8y + 11 = 0• Complete the square for x/y• x² - 4x + 4 + y² + 8y + 16 = -11 + 4 + 16• (x – 2)² + (y + 4)² = 9• YAY! Standard Form!

• B) GRAPH• Plot Center (2,-4)• Radius = 3

WRITE and GRAPH

• A) write the equation of the circle in standard form

• 4x² + 4y² + 36y + 5 = 0• Group the x and y terms• 4x² + 4y² + 36y + 5 = 0• Complete the square for x/y• 4x² + 4(y² + 9y) = -5• 4x² + 4(y² + 9y + 81/4) = -5 + 81• 4x² + 4(y + 9/2)² = 76• x² + (y + 9/2)² = 19• YAY! Standard Form!

• B) GRAPH• Plot Center (0 , -9/2)• Radius = √19 = 4.5

WRITING EQUATIONSWrite the EQ of a circle that has a center of (-5,7) and passes through (7,3)

• Plot your info• Need to find values for h, k, and

r• (h , k) = (-5 , 7)• How do we find r?• Use distance formula with C and

P.

• Plug into formula• (x – h)² + (y – k)² = r²• (x + 5)² + (y – 7)² = (4√10)²• (x + 5)² + (y – 7)² = 160

C = (-5,7)

P = (7,3) radius

104)75()37(Dist 22

Let’s Try OneWrite the EQ of a circle that has endpoints of the diameter at (-4,2) and passes through (4,-6)

A = (-4,2)

B = (4,-6)

radius

• Plot your info• Need to find values for h, k, and

r• How do we find (h,k)?• Use midpoint formula

• (h , k) = (0 , -2)• How do we find r?• Use dist form with C and B.

• Plug into formula• (x – h)² + (y – k)² = r²• (x)² + (y + 2)² = 32

2

62,

2

44C

2432Dist

Hint: Where is the center? How do you find it?

Suppose the equation of a circle is (x – 5)² + (y + 2)² = 9• Write the equation of the new circle given that:

A) The center of the circle moved up 4 spots and left 5:

•(x – 0)² + (y – 2)² = 9 Center moved from (5,-2) (0,2)

B) The center of the circle moved down 3 spots and right 6:

•(x – 11)² + (y + 5)² = 9 Center moved from (5,-2) (11,-5)

Find the center and radius of the circle with equation

(x + 4)2 + (y – 2)2 = 36.

Let‘s Try One

The center of the circle is (–4, 2). The radius is 6.

(x – h)2 + (y – k)2 = r 2 Use the standard form.

(x + 4)2 + (y – 2)2 = 36 Write the equation.

(x – (–4))2 + (y – 2)2 = 62 Rewrite the equation in

standard form.

h = –4 k = 2 r = 6 Find h, k, and r.

Graph (x – 3)2 + (y + 1)2 = 4.

Let’s Try One

(x – h)2 + (y – k)2 = r 2 Find the center and radius of the circle.

(x – 3)2 + (y – (–1))2 = 4

h = 3 k = –1 r 2 = 4, or r = 2

Draw the center (3, –1) and radius 2.Draw a smooth curve.