Verify if the line 2x-y-10=0 is tangent to the circle x^2+y^2-4x+2y=0.

To verify if 2x-y-10 is a tangent to
x^2+y^2-4x+2y=0.

We know that if 2x-y-10 is a tangent to
the circle x^2+y^2-4x+2y=0, then the radius of the circle is equal to the  distance of
the centre C(h,k).

The given circle is written as:
(x-2)^2-2^2+(y+1)^2-1^2 = 0

(x-2)^2+(y+1)^2 =
5.

So the centre is C(2, -1) and radius r= sqrt 5, or r^2 =
5.

Therefore the distance d of C(2, -1) from line 2x-y-10 =
0 is given by:

d= |2*xC- yC -10|/sqrt{2^2+(-1)^2} =
|2*2-1*-1-10|/sqrt(5)

d = |5-10|/sqrt5 = 
sqrt5.

Therefore d = r = sqrt5
.

Therefore the given line is the tangent to the
circle.