Maximum Y in a rotate ellipse with a, b and phi

Question

We have major axis, minor axis and the phi between major axis and y axis in a rotated ellipse. How can we find the maximum y?

Answer 1

It's easy to see that $\phi=0$ gives the maximum $y$. If you think vectorially, the position vector in direction of the major axis is longest. If you take a different vector and then project it, it gets shorter and shorter again.