If the equality is true for all points P & Q on ellipse then letting P(a,0) & Q(-a,0) we will have:
$\displaystyle e\text{:=}\sqrt{1-\frac{b^2}{a^2}}$
$\displaystyle \frac{1}{\text{PF}}+\frac{1}{\text{QF}} =\frac{1}{a-ea}+\frac{1}{a+ea}= \frac{2 a}{b^2}$
In the general case when P & Q are elsewhere on the ellipse. We have:
Move x-y coordinate's origin to focus F...then PF & QF will be:
$\displaystyle \text{PF}\text{:=}a(1-e \text{Cos}[\theta ])$
$\displaystyle \text{QF}\text{:=}a(1-e \text{Cos}[\theta -\pi ])$
$\displaystyle \frac{1}{\text{PF}}+\frac{1}{\text{QF}} = \frac{2}{a-a e^2 \text{Cos}[\theta ]^2} = -\frac{2}{-a+\left(a-\frac{b^2}{a}\right) \text{Cos}[\theta ]^2}$