If the equation of the circle is x^2 + y^2 = r^2 and the equation of the tangent line is y =mx + b, show that the point of tangency is (-r^2m/b , r^2/b)
The slope of the line from (0,0) to (h,k) is k/h
In this case, that is
(r^2/b) / (-r^2m/b) = -1/m
That is, the radius from (0,0) to y=mx+b is perpendicular, as desired.
Now just show that the distance from (0,0) to y=mx+b is r.
Thus, the line is tangent to the circle at that point.