The equation of a line through a point (x1,y1)
with slope m is 

The equation of every line through (4,-1) with slope m is





Usually if a line intersects a parabola, it intersects it in TWO points
However, a tangent line must intersect a parabola in only ONE (not TWO!)
points.  So when we solve the system of the parabola and the line:



we must get only ONE solution.  We set the right sides equal:





To guarantee this has only ONE solution, the discriminant must be 0.

, a=1, b=4-m, c=4m+4.


So we set this discriminant equal to 0:



m=0; m-24=0
       m=24
So the equations of the tangent lines are

 with m=0, and m=24

 and 

 and    <---answers

The two green lines are the tangent lines.  The points of tangency are at the vertex of the parabola (-2,-1) and the point (10,143)



Edwin