Conic Sections and Parametric Equations · Hard problem

For the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$ with focus $F=(3,0)$, there is a vertical line $x=k$ such that every point $P$ of the ellipse satisfies $PF=e\cdot(\text{distance from }P\text{ to the line})$, where $e$ is the eccentricity. Find $k$.