Find all pt of tangency and equation of all the tangent lines?

Question

Find all pt of tangency and equation of all the tangent lines if a line with slope -4 is tangent to the circle x^2+y^2-10x+4y+12=0

Answer 1

Suppose that the equation of the line is y = (-4)x+b .
Plug the eqiation of the line into the equation of the circle
we have x^2 +(-4x+b)^2-10x+4(-4x+b)+12=0 .
17x^2+(-8b-26)x+(b^2+4b+12)=0

Since the intersection of the tangent line and the circle
is only one pt , we have
(-8b-26)^2-68(b^2+4b+12)=0
Solve it to get b . ==> b^2-36b+35=0 , b=1 or 35

Hence the equation of the two tangent line which is tangent to the circle is y= -4x+1 and y=-4x+35 . (Note that there're two such lines )

With it you can find the points , can't you ?