Using the following equation, find the center and radius of the circle. You must show all work and calculations to receive credit. x^2 − 4x + y^2 + 8y = −4

center (2 , -4)

radius = 4

Step-by-step explanation:

center : (4/2 , -8/2) => (2 , -4)

radius = √(2²+4²-4) = √(4+16-4) = √16 = 4

Subtract 11 from both sides of the equation. x 2 + 4 x + y 2 − 8 y = − 11 Complete the square for x 2 + 4 x . Use the form a x 2 + b x + c , to find the values of a , b , and c . a = 1 , b = 4 , c = 0 Consider the vertex form of a parabola. a ( x + d ) 2 + e Substitute the values of a and b into the formula d = b 2 a . d = 4 2 ( 1 ) Cancel the common factor of 4 and 2 . d = 2 Find the value of e using the formula e = c − b 2 4 a . Tap for more steps... e = − 4 Substitute the values of a , d , and e into the vertex form a ( x + d ) 2 + e . ( x + 2 ) 2 − 4 Substitute ( x + 2 ) 2 − 4 for x 2 + 4 x in the equation x 2 + 4 x + y 2 − 8 y = − 11 . ( x + 2 ) 2 − 4 + y 2 − 8 y = − 11 Move − 4 to the right side of the equation by adding 4 to both sides. ( x + 2 ) 2 + y 2 − 8 y = − 11 + 4 Complete the square for y 2 − 8 y . Use the form a x 2 + b x + c , to find the values of a , b , and c . a = 1 , b = − 8 , c = 0 Consider the vertex form of a parabola. a ( x + d ) 2 + e Substitute the values of a and b into the formula d = b 2 a . d = − 8 2 ( 1 ) Simplify the right side.  d = − 4 Find the value of e using the formula e = c − b 2 4 a . e = − 16 Substitute the values of a , d , and e into the vertex form a ( x + d ) 2 + e . ( y − 4 ) 2 − 16 Substitute ( y − 4 ) 2 − 16 for y 2 − 8 y in the equation x 2 + 4 x + y 2 − 8 y = − 11 . ( x + 2 ) 2 + ( y − 4 ) 2 − 16 = − 11 + 4 Move − 16 to the right side of the equation by adding 16 to both sides. ( x + 2 ) 2 + ( y − 4 ) 2 = − 11 + 4 + 16 Simplify − 11 + 4 + 16 . Tap for fewer steps... Add − 11 and 4 . ( x + 2 ) 2 + ( y − 4 ) 2 = − 7 + 16 Add − 7 and 16 . ( x + 2 ) 2 + ( y − 4 ) 2 = 9 This is the form of a circle. Use this form to determine the center and radius of the circle. ( x − h ) 2 + ( y − k ) 2 = r 2 Match the values in this circle to those of the standard form. The variable r represents the radius of the circle, h represents the x-offset from the origin, and k represents the y-offset from origin. r = 3 h = − 2 k = 4 The center of the circle is found at ( h , k ) . Center: ( − 2 , 4 ) These values represent the important values for graphing and analyzing a circle. Center: ( − 2 , 4 ) Radius: 3

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