How do I find the value of k based on the fact that x+k is a tangent to a parametric equation?

Question

I have been given the question of:

A curve has the parametric equations x=2$t^2$ and y=4t. Find the value(s) of k if y=x+k is a tangent to the curve.

Being the first question I've gotten of this nature I have no idea on how to go about it.

Answer 1

So first find cartesian equation of parametric equation: y=4t so y^2=16t^2 then y^2/x=8 so x=y^2/8. also y=x+k so x=y-k. now y^2/8=y-k so y^2-8y+8k=0. it is clear we want one solution for y or else the line will intersect at more than one point and not be tangent. so discriminant =0, (-8)^2-32k=0 so k=2. Final answer y=x+2 only one solution