I don't have the patience to do the details, but here's an outline of an approach:

You need to get the coordinates x = x(t) and y = y(t) of the intersection point R of the two tangents. The coordinates of R obviously define a curve parametrically - hopefully the curve !

To get the intersection point you need the equation of the two tangents. To get these equations, you need the gradient and a known point. The known point is given in each case. You get the gradient from .

I'd use implicit differentiation to get .

At the point , .

So the tangent at this point has equation

.... (1)

At the point , .

So the tangent at this point has equation

.... (2)

Solve equations (1) and (2) simultaneously to get the intersection point R.

I reserve the right for the above outline to contain careless errors and typos.