Find the center and radius of the circle with the given equation.
-$$x ^ { 2 } + y ^ { 2 } + 14 x + 10 y = - 25$$
A) Center $( - 7 , - 5 ) , r = 7$
B) Center $( 5,7 ) , r = 49$
C) Center $( - 5 , - 7 ) , r = 7$
D) Center $( 7,5 ) , r = 49$

Question

Quizplus · Accepted Answer

To find the center and radius of the circle given by the equation $x^2 + y^2 + 14x + 10y = -25$, we complete the squares for $x$ and $y$. The equation can be rewritten as:$$x^2 + 14x + y^2 + 10y = -25$$Completing the square for $x^2 + 14x$, we add and subtract $(\frac{14}{2})^2 = 49$, and for $y^2 + 10y$, we add and subtract $(\frac{10}{2})^2 = 25$, leading to:$$x^2 + 14x + 49 + y^2 + 10y + 25 = -25 + 49 + 25$$$$x^2 + 14x + 49 + y^2 + 10y + 25 = 49$$$$(x + 7)^2 + (y + 5)^2 = 49$$This equation is in the standard form $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius of the circle. Therefore, the center is $(-7, -5)$ and the radius is $\sqrt{49} = 7$.

Find the Center and Radius of the Circle with the Given