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

Options

A. 2
B. 4
C. 6
D. 8

Answer

B

Explanation

Let e be the eccentricity of the ellipse `(x^(2))/(9)+(y^(2))/(5)=1` Then,
`e=sqrt(1-(5)/(9))=(2)/(3)`
So, the coordinates of its foci are (2, 0) and (-2, 0)
`f_(1)=2 and f_(2)=-2`
The coordinates of foci of parabola `P_(1) and P_(2)` are (2,0) and (-4, 0) respectively. Both parabola have their vertices at the origin. So their equation are
`P_(1):y^(2)=8x and p_(2):y^(2)=-16x`
The equation of tangents to `P_(1) and P_(2)` are
`T_(1):y=m_(1)x+(2)/(m_(1)) andT_(2):y=m_(2)x-(4)/(m_(2))`respectively
It is given that `T_(1) and T_(2)` pass through (-4,0) and (2, 0) respectively.
`therefore 0=4m_(1)+(2)/(m_(1))and 0=2m_(2)-(4)/(m_(2))`
`rArr m_(1)^(2)=(1)/(2)and m_(2)^(2)=2`
`therefore (1)/(m_(1)^(2))+m_(2)^(2)=2+2=4`