(1) Let the chord endpoints centered on A be B and C, and connect the center O with one of the endpoints B and OA. Because A is the center and OA is perpendicular to BC, if there is Pythagorean theorem, OA=2√5. When the line segment perpendicular to the X-axis from point A intersects the X-axis at point D, because the coordinate of A is (5,4), then AD=4, and then it is tested by Pythagorean theorem. So the equation of a circle is (x-3) 2+y 2 = 25.

The most important thing in this problem is to find the coordinates of the center of the circle. It's clear after drawing a picture.