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Circles Date: _____________ 9.3 Circles Circles Standard Equation of a Circle (x – h)2 + (y – k)2 = r2 center: (h, k) radius: r Write the equation of the circle that whose center is at (-2,4) and whose radius is 3. ( x h) ( y k ) r 2 2 2 ( x 2) ( y 4) 3 2 2 ( x 2) ( y 4) 9 2 2 2 Write the equation of the circle that whose center is at (-1,-6) and whose radius is 6. ( x h) ( y k ) r 2 2 2 ( x 1) ( y 6) 6 2 2 ( x 1) ( y 6) 36 2 2 2 Write the equation of the circle that whose center is at (0,4) and whose radius is 2. ( x h) ( y k ) r 2 ( x 0) ( y 4) 2 2 2 2 2 2 x ( y 4) 4 2 2 Find the center and radius of the circle. Then write the equation of the circle. y Center = (1,2) Radius = 3 x ( x 1) ( y 2) 3 2 2 ( x 1) ( y 2) 9 2 2 2 Find the center and radius of the circle. Then write the equation of the circle. y Center = (-1,0) Radius = 4 x ( x 1) ( y 0) 4 2 2 ( x 1) y 16 2 2 2 Find the center and radius of the circle. Then graph the circle. ( x 1) ( y 3) 16 2 Center = (-1,3) 2 y Radius = 4 x Find the center and radius of the circle. Then graph the circle. ( x 3) ( y 2) 4 2 Center = (-3,-2) 2 y Radius = 2 x Find the center and radius of the circle. Then graph the circle. ( x 4) ( y 1) 25 2 Center = (4,1) 2 y Radius = 5 x Complete the square. x2 + 10x 1. 2. 3. 4. 5. 10 2 =5 52 = 25 x2 + 10x + 25 (x + 5)(x + 5) (x + 5)2 Complete the square. x2 – 20x 1. 2. 3. 4. 5. -20 2 = -10 (-10)2 = 100 x2 – 20x + 100 (x – 10)(x – 10) (x – 10)2 Completing the Square to Write Standard Equations of Circles 1. Add/subtract to move the constant term to the other side of the equation. 2. Rearrange the terms so that the x’s and y’s are together. 3. Complete the square for the x’s and y’s. 4. Make sure that anything you added to the one side of the equation is added to the other side as well. Write the equation of the circle in standard form. Then find the center and radius of the circle. x2 + y2 + 4x – 6y – 3 = 0 x2 + y2 + 4x – 6y = 3 x2 + 4x + y2 – 6y = 3 4 + y2 – 6y + ___ 9 = 3+ 4+ 9 x2 + 4x + ___ x2 + 4x + 4 + y2 – 6y + 9 = 16 (x + 2)2 + (y – 3)2 = 16 Center = (-2, 3) Radius = 4 Write the equation of the circle in standard form. Then find the center and radius of the circle. x2 + y2 – 12x – 2y – 8 = 0 x2 + y2 – 12x – 2y = 8 x2 – 12x + y2 – 2y = 8 36 + y2 – 2y + ___ 1 = 8 +36 +1 x2 – 12x + ___ x2 – 12x + 36 + y2 – 2y + 1 = 45 (x – 6)2 + (y – 1)2 = 45 Center = (6, 1) Radius = √45 ≈ 6.7 Write the equation of the circle in standard form. Then find the center and radius of the circle. x2 + y2 – 10x + 4y + 17 = 0 x2 + y2 – 10x + 4y = -17 x2 – 10x + y2 + 4y = -17 25 + y2 + 4y + ___ 4 = -17+25 +4 x2 – 10x + ___ x2 – 10x + 25 + y2 + 4y + 4 = 12 (x – 5)2 + (y + 2)2 = 12 Center = (5, -2) Radius = √12 ≈ 3.5 Write the equation of the circle that passes through the given point and has a center at the origin. y (-5, -12) d x2 x1 d 5 0 d 5 d 25 144 d 169 2 2 d 13 2 y2 y1 2 12 0 12 2 x 2 x y 2 2 169