To find the equation of the circle, we need to determine its center coordinates (h, k) and its radius r.

Given that the circle intersects the x-axis at points P(13,0) and Q(27,0) and intersects the y-axis at point R(0,9), we can determine the center coordinates (h, k) as the midpoint of the line segment connecting points P and Q, and the radius r as the distance from the center to any of the points on the circle.

Step 1: Determine the center coordinates (h, k):
The x-coordinate of the center is the average of the x-coordinates of P and Q:
h = (13 + 27)/2 = 40/2 = 20

The y-coordinate of the center is the same as the y-coordinate of point R:
k = 9

Therefore, the center coordinates of the circle are (h, k) = (20, 9).

Step 2: Determine the radius r:
The radius r is the distance between the center (20, 9) and any point on the circle, such as point P or Q.

Using the distance formula:
r = sqrt((x2 - x1)^2 + (y2 - y1)^2)

For point P(13,0):
r = sqrt((13 - 20)^2 + (0 - 9)^2) = sqrt((-7)^2 + (-9)^2) = sqrt(49 + 81) = sqrt(130)

Therefore, the radius of the circle is r = sqrt(130).

Step 3: Write the equation of the circle:
The equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2

Plugging in the values, we have:
(x - 20)^2 + (y - 9)^2 = (sqrt(130))^2
(x - 20)^2 + (y - 9)^2 = 130

Therefore, the equation of the circle is (x - 20)^2 + (y - 9)^2 = 130.