Suppose that the foci of the ellipse `(x^2)/9+(y^2)/5=1` are `(f_1,0)a n d(f_2,0)` where `f_1>0a n df_2<0.` Let `P_1a n dP_2` be two parabolas with a common vertex at (0, 0) and with foci at `(f_1 .0)a n d` (2f_2 , 0), respectively. Let`T_1` be a tangent to `P_1` which passes through `(2f_2,0)` and `T_2` be a tangents 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 `(1/(m_1^ 2)+m_2^ 2)` is

Explanation

Ellipse is `(x^(2))/(9)+(y^(2))/(5)=1`
`:. A^(2)e^(2)=a^(2)-b^(2)=9-5=4`
`:.` foci are (2,0) and (-2,0)
Parbola with vertex O(0,0) and focus `(f_(1),0) or (2,0) or (2,0) "is" y^(2)=8x`.
Equation of tangent having slop `m_(1)` is `u=m_(1)x+(2)(m_(1))` ltbr It passes through `(2f_(2)0) or (-4,0)`
`:. 0-4m_(1)+(2)/(m_(1)) rarr m_(1)=+(1)/(sqrt(2))`
Parabola with vertex O(0,0) and focus `(2f_(2),0) or (-4,0)` is `y^(2)=-16x`
Equation of tangent having slope `m_(2)` is `y=m_(2)x-(4)/(m_(2))`
It passes through `(f_(1),0) or (2,0)`
`:. 0=2m_(2)-(4)/(m_(2)) rArr m_(2)=+sqrt(2)`
`rArr (1)/(m_(1)^(2))+m_(2)^(2)=2+2=4`