JEE-Advanced 2015 (24/40)

Suppose that the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$ 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 find the value of $\left( \frac{1}{m_1^2}+m_2^2 \right)$