Question

which equation represents a circle centered at (-2,-4) and passing through the point (4,-8)?
a. $(x - 2)^2+(y - 4)^2 = 10$
b. $(x + 2)^2+(y + 4)^2 = 52$
c. $(x + 2)^2+(y + 4)^2 = 10$
d. $(x - 2)^2+(y - 4)^2 = 52$

Explanation:

Step1: Recall the circle - equation formula

The standard form of the 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 the radius. Given the center $(h,k)=(-2,-4)$, the equation of the circle is $(x + 2)^2+(y + 4)^2=r^2$.

Step2: Calculate the radius

The circle passes through the point $(x_1,y_1)=(4,-8)$. The radius $r$ is the distance between the center $(h,k)=(-2,-4)$ and the point $(x_1,y_1)=(4,-8)$. Using the distance formula $d=\sqrt{(x_1 - h)^2+(y_1 - k)^2}$, we have $r=\sqrt{(4+2)^2+(-8 + 4)^2}=\sqrt{6^2+(-4)^2}=\sqrt{36 + 16}=\sqrt{52}$. Then $r^2 = 52$.

Answer:

B. $(x + 2)^2+(y + 4)^2=52$