Find the equation of the circle that satisfies the following conditions. Express the final equation in the form x ² + y ² + Dx + Ey + F = 0. Tangent to the x axis, a radius of length 7, and abscissa of center is -8 A)x ² + y ² - 16 x - 14 y + 64 = 0 and x ² + y ² - 16 x + 14 y + 64 = 0 B)x ² + y ² + 16 x - 14 y + 64 = 0 C)x ² + y ² + 16 x - 14 y + 64 = 0 and x ² + y ² + 16 x + 14 y + 64 = 0 D)x ² + y ² + 16 x + 14 y + 64 = 0 E)x ² + y ² - 16 x + 14 y + 64 = 0

Question

Quizplus · Accepted Answer

The general equation of a circle is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is its radius. Given that the circle is tangent to the x-axis, its center must be $7$ units away from the x-axis, since the radius is $7$. The abscissa of the center is $-8$, so $h = -8$. Since it's tangent to the x-axis, the ordinate $k$ can be either $7$ or $-7$, making the center $(-8, 7)$ or $(-8, -7)$. Substituting $h = -8$, $k = 7$ or $k = -7$, and $r = 7$ into the circle equation gives two equations: $(x + 8)^2 + (y - 7)^2 = 49$ and $(x + 8)^2 + (y + 7)^2 = 49$. Expanding and simplifying these into the given form yields the two equations in choice C.

Find the Equation of the Circle That Satisfies the Following