We know the equation for a circle is given by
( x-h) ^2 + ( y-k) ^2 = r^2 where ( h,k) is the center and r is the radius
We know the center of the circle is at ( 4,-3)
( x-4) ^2 + ( y--3) ^2 = r^2
( x-4) ^2 + ( y+3) ^2 = r^2
Now we need to find the radius which is the distance from the center to the point (5,1)
r = sqrt( ( x2-x1)^2 + ( y2 - y1) ^2 )
= sqrt( ( 5 - 4) ^2 + ( 1 - -3) ^2)
= sqrt( ( 1^2 + ( ( 1+3) ^2)
= sqrt( 1 + 4^2)
= sqrt( 1+16)
= sqrt( 17)
Squaring each side
r^2 = 17
( x-4) ^2 + ( y+3) ^2 = 17