Considering the ellipse $4x^2+9y^2-36=0$, which can be expressed in the standard form of an ellipse as $\frac{x^2}{3^2}+\frac{y^2}{2^2}=1$.
PROBLEM I:
A random chord through the origin of the ellipse is constructed. What is the expected length of that chord?
Here is almost a similar problem.
PROBLEM II:
A random chord (not necessary through the origin) is constructed. What is the expected length of that chord?
PROBLEM III:
A random chord is constructed such that the area bounded by the chord and the smaller arc of the ellipse is exactly $\frac{3}{2}\pi$ units$^2$. What is the expected length of the chord.
For me, these three problems are easy to understand and difficult, currently, to solve.
Providing me the useful geometry and probability facts and theorems will help me to solve them and to show my work, then you can comment on my answers. For now, I have not attempted any.
What I know for now, only the area of that ellipse is $3 \times 2 \times \pi=6\pi$ units$^2$.
Your help would be really appreciated. THANKS!