Solving Two Parabola Problems: Proving & Finding Values

[varunKanpur]

I am not able to solve the following problem

#1) Prove that the normal to parabola y²=4ax at (am²,-2am) intersects the parabola again at an angle tan-1(m/2)

What I am thinking is to solve the equation of parabola and equation of normal y=mx-am^-3-2am simultaneously and at that point I will find the slope of tangent and will get the angle between tangent and normal. The problem is that answer is not coming.

#2) For what values of a will the tangents drawn to parabola y²=4ax from a point , not on the y-axis, will be normals to the parabola x²=4y?

I have no idea on how to solve this question
Thanks in Advance

 

[Simon Bridge]

Start with the first one to begin with.

Please show your working?
Show each step with your reasoning.
It can hep to reverse the roles of the x and y axes.

The key to the second one is to rewrite the problem statement in maths.
i.e. a point not on the y-axis is point $p=(p_x,p_y): p_x\neq 0$

 

[varunKanpur]

I got the #1 problem, I was making it more lengthy.

 

[Simon Bridge]

OK - so what about #2?
Same suggestions.