whats the simplest way to find this circle's center if known its tangent line

Question

the circle has a tangent line $y = 2x + 1$ at $(2,5)$ and its center on the line $y = 9 - x$. If that's circle intersect the $x$ -axis at $x_1, x_2$ what's $x_1 + x_2$ ?

i understand than $x_1 + x_2 = 2x_0$ when $x_0$ is the circle's center. we can use $(x-a)^2 + (y-b)^2 = r^2$ when $b = 9-a$ can we use different method? using $y=mx \pm r\sqrt{m^2+1}$ seems complicated.

Answer 1

The perpendicular line to $y=2x+1$ at $P: (2,5)$ intersects the line $y=9-x$ in $C: (6,3)$ which is the center of your circle. Now $CP$ is a radius, so the equation of the circle is $(x-6)^2+(y-3)^2=20.$ Can you take it from here?