Height of a rotated ellipse

Question

If I have an ellipse, it is easy to find its height, twice the length of the major axis. But if the ellipse is rotated a certain number of degrees, how do you find the vertical height from top to bottom?

Answer 1

Hint: The height is expressed by y. Extract y in terms of x, using the equation of the rotated ellipse, and then remember what you were taught in school about finding the maximum and minimum of a function...

Answer 2

Hint:

An ellipse of center in the origin and the axis rotated by an angle $\theta$ has equation: $$ \frac{(x\cos \theta+y\sin \theta)^2}{a^2}+\frac{(y\cos \theta-x\sin \theta)^2}{b^2}=1 $$ that can be write as: $$ Ax^2+Bxy+Cy^2=1 $$ with $B^2-4AC<0$. From this find: $$ y=\dfrac{-Bx\pm\sqrt{B^2x^2-4C(Ax^2-1)}}{2C} $$ and you have two equation of two semi-ellipses. Now find the maximum and minimum of these functions and the difference of the ordinates of these points is the searched height.