JEE Advanced 2015 · Paper 2, Mathematics question 44

Suppose that the foci of the ellipse $\dfrac{x^2}9+\dfrac{y^2}5=1$ are $(f_1,0)$ and $(f_2,0)$ where $f_1>0$ and $f_2<0$. Let $P_1$ and $P_2$ be two parabolas with a common vertex at $(0,0)$ and with foci at $(f_1,0)$ and $(2f_2,0)$, respectively. Let $T_1$ be a tangent to $P_1$ which passes through $(2f_2,0)$ and $T_2$ be a tangent to $P_2$ which passes through $(f_1,0)$. If $m_1$ is the slope of $T_1$ and $m_2$ is the slope of $T_2$, then the value of $\left(\dfrac1{m_1^2}+m_2^2\right)$ is