Relationships between the eccentricities of two hyperbolas whose latusrectums form a rectangle

Question

The four Latusrectum of two hyperbolas forms four distinct sides of a rectangle. If $e,E$ are their eccentricities, then which of the following are true?

1. $eE\geq 2$

2. $\left(e-\dfrac{1}{e}\right)\left(E-\dfrac{1}{E}\right)=1$

3. $eE\geq 3$

4. both hyperbolas can never have same eccentricities

This is a question in a test of my student. We knew the answers 1), 2) are correct. But couldn't figure out why?

I have tried to make figures, but converting into algebraic expressions is difficult.

It would be a great pleasure if someone helps!