What should i do to prove that the line y=2x-10 is tangent to circle x^2-4x+y^2+2y=0?

To prove that y = 2x - 10 is tangent to the circle we find the
center and radius of the circle and show that the distance of the center from the line is equal
to the radius.

The circle: x^2 - 4x + y^2 + 2y =
0

=> x^2 - 4x + 4 + y^2 + 2y + 1 =
-5

=> (x - 2)^2 + (y + 1)^2 = (sqrt
5)^2

The center of the circle is (2, -1) and the radius is sqrt
5

The distance of the point (2, -1) from y = 2x - 10
is

|2*2 + 1 - 10|/sqrt (4 +
1)

=> 5/sqrt 5

=> sqrt
5

We see that the point (2, -1) does lie at a distance equal to
(sqrt 5) from the line 2x - y - 10 = 0

This proves
that y = 2x - 10 is tangent to circle x^2 - 4x + y^2 + 2y =
0