Identify the equation of the circle that has its center at (7, - 24) and passes through the origin.

The equation of this line would be (x - 7) ^2 + (y + 24) ^2 = 625

This is because of the base equation for circles, which is:

(x - x1) ^2 + (y - y1) ^2 = r^2

In which, x1 is the x-coordinate to the center, y1 is the y-coordinate to the center and r is the radius.

So we start by putting in out known values.

(x - 7) ^2 + (y + 24) ^2 = r^2

However, we still don't know the r value. To find it, we must use the Pythagorean Theorem to find the distance between the center and the origin.

a^2 + b^2 = c^2

7^2 + 24^2 = c^2

49 + 576 = c^2

625 = c^2

25 = c

So we know the radius must be 25. So we can plug that into what we already had.

(x - 7) ^2 + (y + 24) ^2 = 25^2

(x - 7) ^2 + (y + 24) ^2 = 625