First, to find the equation of a circle, we need to know its radius and the coordinates of its center. In this case, the center of the circle (h,k) is given as (-9,1).
The general equation for a circle is given by (x - h)^2 + (y - k)^2 = r^2 , where r is the radius of the circle.
Given a point on the circle (13,6) (x,y), we can use the distance formula to calculate the radius. The distance formula is given by the square root of ((x-h)^2 + (y-k)^2), or in other words the square of the radius (r^2) is ((x-h)^2 + (y-k)^2). Substituting the given values, we have r^2 = ((13 – (-9))^2 + (6 - 1)^2), which simplifies to r^2 = 509.
Now we have all the parameters needed for the equation of the circle. The equation of the circle thus becomes: (x - (-9))^2 + (y - 1)^2 = 509 or alternatively (x +9)^2 + (y - 1)^2 = 509.