Suppose that the foci of the ellipse `(x^(2))/(9)+(y^(2))/(5)=1` are `(f_(1),0) and (f_(2),0)` where `f_(1)gt0 and f_(2)lt0` Let `P_(1) and P_(2)` be two parabola with a c ommon vertex at (0, 0) and with foci at `(f_(1),0) and (2f_(z),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 `((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`