Hyperbolic functions problem

Question

If $p^2\sinh x+q^2\cosh x = r^2$ has at least one root, how do I show that $r^4 > p^4-q^4?$

Answer 1

Your equation is equivalent to:

$$(p^2+q^2)e^x+(p^2-q^2)e^{-x}=2r^2$$

or:

$$(p^2+q^2)e^{2x}-2r^2e^x+(p^2-q^2)=0\tag{1}$$

This is a quadratic equation with respect to $t=e^x$. If (1) has at least one root, it means that there has to be one positive solution of (1). And it is possible if:

$$D=(2r^2)^2-4(p^2+q^2)(p^2-q^2)\ge 0$$

...which leads directly to:

$$r^4\ge p^4-r^4$$