Question

find the equation of circle
if the line 2x - y +1 = 0 touches the circle at the point (2,5) and the center of the circle lies on the line x+y -9 =0

Answer 1

If the line touches the circle, then the line from (2,5) to the center is perpendicular to the line y=2x+1, so it has slope -1/2.

So, the line through (2,5) with slope -1/2 is

y-5 = -1/2 (x-2)
y = -1/2 x + 6

This line passes through the center of the circle, so

-1/2 x + 6 = 9-x
x=6

(6,3) is the center of the circle.
The radius is the distance from (6,3) to (2,5) = √20

The circle is thus

(x-6)^2 + (y-3)^2 = 20

Just as a final check:
The distance from (6,3) to the line 2x-y+1 = 0 is |2*6 - 3 + 1|/√5 = 10/√5 = 2√5 = √20