Circles

Find the missing value to complete the square. 6.x 2 – 2x +7. x 2 + 4x +8. x 2 – 6x + Circles – Warm Up Find the missing value to complete the square. 6.x 2 – 2x +7. x 2 + 4x +8. x 2 – 6x + Simplify

Solutions 6.x 2 – 2x + ; c = = – = (–1) 2 = 1 7.x 2 + 4x + ; c = = = 2 2 = 4 8.x 2 – 6x + ; c = = – = (–3) 2 = 9 b2b2 b2b2 b2b b2b2 b2b2 b2b2 b2b = = = 4  5 = = 16  3 = = 36  2 = 6 2

CIRCLE TERMS EQUATION FORM CENTER RADIUS MIDPOINT FORMULA DISTANCE FORMULA (x – h)² + (y – k)² = r² (h, k ) r 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 = r 2 Use the standard form of the equation of a circle. (x – 3) 2 + (y – (–2)) 2 = 3 2 Substitute 3 for h, –2 for k, and 3 for r. (x – 3) 2 + (y + 2) 2 = 9Simplify. 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 x 2 + y 2 = 16 two units right and one unit down. Circles (x – 2) 2 + (y – (–1)) 2 = 16Substitute 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 = 16Simplify. 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 y² + 8y + 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) = 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) P = (7,3) radius

Let’s Try One Write 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 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 = 36Write the equation. (x – (–4)) 2 + (y – 2) 2 = 6 2 Rewrite the equation in standard form. h = –4 k = 2 r = 6Find 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.