Why try to eliminate the parameter? There is a reason why folks got into parametric equations. Sometimes, you can do more things.

Find the general slope.

You have to know where that came from.

Now, write the equations of the tangent lines.

Don't forget to substitute the right values in dy/dx.

Solve system, for x and y.

I get x = a(cos(t)-sin(t)) and y = b(cos(t)+sin(t))

One last thing. Is that point on the new ellipse?

This is a great review problem. There is a ton of mathematics in there. Understand this problem and you will have some good information for the future.