Coordinate geometry help?

Question

Given that y-axis is a tangent to the circle with radius 5 units. If the cirlce passes through the point (1,-4), find the possible equation of the circle.

Answer 1

Let (0,b) be the point of tangency. Then a radius drawn to (0,b) from the center of the circle is a horizontal line segment of length 5 (because a radius drawn to the point of tangency is perpendicular to the tangent line). So the other endpoint of the segment, which is the center of the circle, is (5,b)

The equation of the circle is

(x-5)² + (y-b)² = 25

The point (1,-4) is on the circle; therefore, its coordinates satisfy the equation of the circle:

(1-5)² + (-4-b)² = 25

Solve for b.