10 3 Circles Circles Warm Up Simplify 1

10. 3 - Circles

Circles – Warm Up Simplify. 1. 16 2. 49 3. 20 4. 48 5. Find the missing value to complete the square. 6. x 2 – 2 x + 7. x 2 + 4 x + 8. x 2 – 6 x + 72

Solutions 1. 16 = 4 2. 49 = 7 3. 20 = 4 5 = 2 5 4. 48 = 16 3 = 4 3 5. 72 = 36 2 = 6 2 6. x 2 – 2 x + ; c = b 2 = 2 2 – 2 2 = (– 1)2 = 1 2 7. x 2 + 4 x + ; c = b 2 2 = 8. x 2 – 6 x + ; c = b 2 2 2 = – 6 = (– 3)2 = 9 4 2 = 22 = 4 2

CIRCLE TERMS r C=(h , k) Definition: A circle is an infinite number of points a set distance away from a center EQUATION FORM (x – h)² + (y – k)² = r² CENTER (h, k ) RADIUS r MIDPOINT FORMULA DISTANCE FORMULA

Circles Write an equation of a circle with center (3, – 2) and radius 3. (x – h)2 + (y – k)2 = r 2 Use the standard form of the equation of a circle. Substitute 3 for h, – 2 for k, and 3 for r. Simplify. (x – 3)2 + (y – (– 2))2 = 32 (x – 3)2 + (y + 2)2 = 9 Check: Solve the equation for y and enter both functions into your graphing calculator. (x – 3)2 + (y + 2)2 = 9 – (x – 3)2 y + 2 = ± y = – 2 ± 9 – (x – 3)2

Circles Write an equation for the translation of x 2 + y 2 = 16 two units right and one unit down. (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 Substitute 2 for h, – 1 for k, and 16 for r 2. (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 - 4 x + 8 y + 11 = 0 • Group the x and y terms • x² - 4 x + y² + 8 y + 11 = 0 • Complete the square for x/y • x² - 4 x + 4 + y² + 8 y + 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 • 4 x² + 4 y² + 36 y + 5 = 0 • Group the x and y terms • 4 x² + 4 y² + 36 y + 5 = 0 • Complete the square for x/y • 4 x² + 4(y² + 9 y) = -5 • 4 x² + 4(y² + 9 y + 81/4) = -5 + 81 • 4 x² + 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 EQUATIONS Write 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) rad ius P = (7, 3)

Let’s Try One Write the EQ of a circle that has endpoints of the diameter at (-4, 2) and passes through (4, -6) • Plot your info • Need to find values for h, k, and r • How do we find (h, k)? • Use midpoint formula us • Plug into formula • (x – h)² + (y – k)² = r² • (x)² + (y + 2)² = 32 di ra • (h , k) = (0 , -2) • How do we find r? • Use dist form with C and B. A = (-4, 2) B = (4, -6) 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)

Let‘s Try One Find the center and radius of the circle with equation (x + 4)2 + (y – 2)2 = 36. (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 Find h, k, and r. k = 2 r = 6 The center of the circle is (– 4, 2). The radius is 6.

Let’s Try One Graph (x – 3)2 + (y + 1)2 = 4. (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.