Suppose that the foci of the ellipse $\frac{x^2}{9}+ \frac{y^2}{5}=1$ are $\left(f_1, 0\right) and (f_2, 0)$ 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 $\left(f_1, 0\right)$ and $\left(2f_2, 0\right),$ respectively. Let $T_1$ be a tangent to $P_1$ which passes through $\left(2f_2, 0\right)$ 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 $\left(\frac{1}{m_1^2}+ m_2^2\right)$ is

Answer $Q.39, Q.40$ and $Q.41$ by appropriately matching the information given in the three columns of the following table.

If the tangent to a suitable conic (List I) at $\left(\sqrt{3},\frac{1}{2}\right)$ is found to be $\sqrt{3}x+2y=4,$ then which of the following  options is the only CORRECT combination?

Statement $1:$ An equation of a common tangent to the parabola $y^2=16\sqrt{3}x$ and the ellipse $2x^2+y^2=4$ is $y=2x+2\sqrt{3}$.

Statement $2:$ If the line $y=mx+\frac{4\sqrt{3}}{m}, \left(m\ne 0\right)$ is a common tangent to the parabola ${\mathrm{y}}^2=16\sqrt{3}\mathrm{x}$ and the ellipse $2x^2+y^2=4,$ then $m$ satisfies $m^4+2m^2=24$.