Find the center (h, k) and radius r of the circle with the given equation.
-$$x ^ { 2 } + y ^ { 2 } + 18 x - 8 y + 97 = 16$$
A) $( \mathrm { h } , \mathrm { k } ) = ( 9 , - 4 ) ; \mathrm { r } = 16$
B) $( \mathrm { h } , \mathrm { k } ) = ( 4 , - 9 ) ; \mathrm { r } = 4$
C) $( h , k ) = ( - 4,9 ) ; r = 16$
D) $( h , k ) = ( - 9,4 ) ; r = 4$

Question

Quizplus · Accepted Answer

To find the center and radius of the circle given by the equation $x^2 + y^2 + 18x - 8y + 97 = 16$, we complete the square for both $x$ and $y$. The general form of a circle's equation is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.Rearrange the equation: $x^2 + 18x + y^2 - 8y = -81$.Complete the square for $x$ and $y$:- For $x^2 + 18x$, add and subtract $(\frac{18}{2})^2 = 81$, resulting in $x^2 + 18x + 81$.- For $y^2 - 8y$, add and subtract $(\frac{8}{2})^2 = 16$, resulting in $y^2 - 8y + 16$.The equation becomes $x^2 + 18x + 81 + y^2 - 8y + 16 = -81 + 81 + 16$, which simplifies to $(x + 9)^2 + (y - 4)^2 = 16$.Thus, the center $(h, k) = (-9, 4)$ and the radius $r = \sqrt{16} = 4$.

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