Answer
The center is $(1,0)$
Work Step by Step
The line $x - 2y + 4 = 0$ is tangent to a circle at (0, 2).
Slope of this tangent = $\frac{1}{2}$
So slope of radius from point (0,2) on the circle = -2
Equation of that radius line => $y - 2 = -2(x-0)$
=> $\frac{y}{2}= -x + 1$ - - - (i)
The line $y = 2x - 7$ is tangent to the same circle at (3, - 1).
Slope of this tangent = $2$
So slope of radius from point (3, - 1) on the circle = $\frac{-1}{2}$
Equation of that radius line => $y - (-1) = \frac{-1}{2}(x-3)$
=> $2y = -x + 1$ - - - (ii)
Find the center of the circle
Centre will be intersection of two radii lines
Solving (i) and (ii) we get
x = 1 and y = 0
So the center is $(1,0)$
