If `P(x ,y)` is any point on the ellipse `16 x^2+25 y^2=400` and `f_1=(3,0)F_2=(-3,0)` , then find the value of `P F_1+P F_2dot`

Options

A. 8
B. 6
C. 10
D. 12

Answer

C

Explanation

We have,
`16x^(2) + 25y^(2) = 400`
`rArr x^(2)/25 + y^(2)/16 = 1` or `x^(2)/a^(2) + y^(2)/b^(2) = 1`, where `a^(2) = 25 and b^(2) = 16`.
This equation represents an ellipse with eccentricity e given by
`e^(2) = 1 - b^(2)/a^(2) = 1 - 16/25 = 9/25 rArr e = 3//5`.
So, the coordinates of the foci are `(pm ae, 0)` i.e. (3, 0) and (-3, 0).
Thus, `F_(1) and F_(2)` are the foci of the ellipse
Since, the sum of the focal distances of a point on an ellipse is equal to its major axis.
`therefore PF_(1) + PF_(2) = 2a = 10.`