Circles Students will be able to transform an equation of a circle in standard form to center, radius form by using the complete the square method.

Circles – Warm Up Simplify. 1. 16 2. 49 3. 20 4. 48 5. 72 1. 16 2. 49 3. 20 4. 48 5. 72 Find the missing value to complete the square. 6. x2 – 2x + 7. x2 + 4x + 8. x2 – 6x + Find the missing value to complete the square. 6. x2 – 2x + 7. x2 + 4x + 8. x2 – 6x +

Solutions 6. x2 – 2x + ; c = = – = (–1)2 = 1 7. x2 + 4x + ; c = = = 22 = 4 8. x2 – 6x + ; c = = – = (–3)2 = 9 b 2 4 6 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 Definition: A circle is an infinite number of points a set distance away from a center EQUATION FORM CENTER RADIUS MIDPOINT FORMULA DISTANCE r (x – h)² + (y – k)² = r² (h, k ) C=(h , k) r

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

Circles Write an equation for the translation of x2 + y2 = 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.

Circles Write an equation for the translation of x2 + y2 = 16 two units right and one unit down. (x – h)2 + (y – k)2 = r 2 the equation of a circle. (x – 2)2 + (y + 1)2 = 16 The equation is (x – 2)2 + (y + 1)2 = 16. Now multiply it out for standard form:

Circles Write an equation for the translation of x2 + y2 = 16 two units right and one unit down. (x – h)2 + (y – k)2 = r 2 the equation of a circle. (x – 2)2 + (y + 1)2 = 16

WRITE the equation of a circle in center, radius form Group the x and y terms Move the constant term Complete the square for x, y Write each variable as a square Ex: x² + y² - 4x + 8y + 11 = 0

WRITE and GRAPH x² + y² - 4x + 8y + 11 = 0 Group the x and y terms Given 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 B) GRAPH Plot Center (2,-4) Radius = 3

WRITE and then GRAPH 4x² + 4y² + 36y + 5 = 0 Group terms, move constant, factor out the coefficient. Complete the square for x,y Divide both sides by coefficient

WRITE and then GRAPH This time factor out the coefficients: 4x² + 4y² + 36y + 5 = 0 Group terms, move constant, factor out the coefficient. 4(x²) + 4(y² + 9y + __ ) = -5 + ___ Notice that x is already done!

WRITE and GRAPH 4x² + 4y² + 36y + 5 = 0 Complete the square for y: A) write the equation of the circle in standard form 4x² + 4y² + 36y + 5 = 0 Complete the square for y: 4(x²) + 4(y² + 9y+__) = -5 +__ 4(x²)+4(y² + 9y + 81/4) =-5+81 4(x²) + 4(y + 9/2)² = 76 Divide by 4: x² + (y + 9/2)² = 19 Center (0 , -9/2) Radius =

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, r (h , k) = ( , ) How do we find r? Use distance form. for C and P. Plug into circle formula (x – h)² + (y – k)² = r² C = (-5,7) P = (7,3) radius

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, r (h , k) = (-5 , 7) How do we find r? Use distance form. for C and P. Plug into circle formula (x – h)² + (y – k)² = r² (x + 5)² + (y – 7)² = 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) Plot your info Need to find values for h, k, r How do we find (h,k)? Use midpoint formula (h , k) = How do we find r? Use dist form with C and B. Plug into formula (x – h)² + (y – k)² = r² A = (-4,2) radius B = (4,-6) Hint: Where is the center? How do you find it?

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, 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 A = (-4,2) radius 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 k = 2 r = 6 Find h, k, and r. 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.

Solving linear quadratic systems Do now: page 182 # 37

Solving a circle-line system Plot the line b=-2 m = -1 Plot the center and count in four directions for the radius Find the points of intersection y=-x -2 x² + (y + 2)² = 32 Y=-x-2

Solving a circle-line system Plot the line b=-2 m = -1 Plot the center and count in four directions for the radius Find the points of intersection y=-x -2 x² + (y + 2)² = 32 (-4,2) Y=-x-2 (4,-6)

Solving a circle-line system By algebraic method: substitute –x-2 for y into second equation y=-x -2 x² + (y + 2)² = 32

Solving a circle-line system By algebraic method: substitute –x-2 for y into second equation y=-x -2 x² + (y + 2)² = 32 x² + (-x - 2+ 2)² = 32 x² + ( -x)² = 32 2x2=32 X2=16 X=4, x=-4 solve for y: Y=-4-2 y=4-2 Y=-6 (4,-6) y=2 (-4, 2)

Lets solve this one both ways

Lets solve this one both ways