To write the equation of a circle we need to know the center and the radius. We are given the center explicitly, and we can deduce the radius: if the circumference passes through the origin, it means that the origin belong to the circumference.
But all points belonging to the circumference have the same distance from the center: the radius! So, the radius of this circumference is the distance between (7, -24) and the origin, which we can compute with the usual formula
[tex] d(A,B) = \sqrt{(A_x-B_x)^2+(A_y-B_y)^2} [/tex]
which in this case becomes
[tex] r = d((7, -24),(0,0) = \sqrt{(7-0)^2+(-24-0)^2} = \sqrt{49+576} = \sqrt{625} = 25 [/tex]
Now that we know the center and the radius of the circle, we can write its equation. In general, given the center C = (h,k) and the radius r, the equation is
[tex] (x-h)^2 + (y-k)^2 = r^2 [/tex]
which in this case becomes
[tex] (x-7)^2 + (y+24)^2 = 625 [/tex]
