Find the center (h, k) and radius r of the circle with the given equation.
-$$x ^ { 2 } + y ^ { 2 } - 4 x + 8 y = 29$$
A) $( \mathrm { h } , \mathrm { k } ) = ( 4 , - 2 ) ; \mathrm { r } = 49$
B) $( \mathrm { h } , \mathrm { k } ) = ( 2 , - 4 ) ; \mathrm { r } = 7$
C) $( \mathrm { h } , \mathrm { k } ) = ( - 2,4 ) ; \mathrm { r } = 49$
D) $( \mathrm { h } , \mathrm { k } ) = ( - 4,2 ) ; \mathrm { r } = 7$

Question

Quizplus · Accepted Answer

To find the center and radius, we complete the square for both $x$ and $y$. The given equation is $x^2 + y^2 - 4x + 8y = 29$. Completing the square:- For $x^2 - 4x$, add and subtract $(\frac{4}{2})^2 = 4$.- For $y^2 + 8y$, add and subtract $(\frac{8}{2})^2 = 16$.So, the equation becomes $x^2 - 4x + 4 + y^2 + 8y + 16 = 29 + 4 + 16$, which simplifies to $(x - 2)^2 + (y + 4)^2 = 49$. The center $(h, k) = (2, -4)$ and the radius $r = \sqrt{49} = 7$.

Find the Center (H, K) and Radius R of the Circle