Question

If (− 4, −8) and (−10, −12) are the endpoints of a diameter of a circle, what is the equation of the circle?

A) (x + 7)^2 + (y + 10)^2 = 13
B) (x + 7)^2 + (y − 10)^2 = 12
C) (x − 7)^2 + (y − 10)^2 = 169
D) (x − 13)^2 + (y − 10)^2 = 13

Answer 1

Correct option: A

(x + 7)^2 + (y + 10)^2 = 13

Explanation:

First we need to find the center of the circle. We can find it calculating the midpoint of the diameter.

the x-coordinates of the diameter are -4 and -10, so the midpoint is:

(-4 -10)/2 = -7

the y-coordinates of the diameter are --8 and -12, so the midpoint is:

(-8 -12)/2 = -10

Now we need to find the radius of the circle, so we find the diameter and then find half of it.

The lenght of the diameter is the distance of the endpoints:

D = sqrt((-4+10)^2 + (-8+12)^2) = 7.21

radius = 7.21/2 = 3.605

The generic equation of the circle is:

(x - xc)^2 + (y - yc)^2 = r^2

So we have:

(x + 7)^2 + (y + 10)^2 = 13

Correct option: A