Hello arzeThanks for showing us all your detailed working. I think you are right, and the stated answer is wrong. Here's why:

The normal must (obviously) pass through the point with parameter . So when and the value of is

So your equation contains the original point, has the required gradient and is therefore the correct equation of the normal.

As far as the second part is concerned, you may find the following idea useful.

If this line is also normal at a point with parameter , then the gradients at and are equal. So, simplifying your expression for ,

Grandad