If parabolas `y^2=lambdax` and `25[(x-3)^2+(y+2)^2]=(3x-4y-2)^2` are equal, then the value of `lambda` is (a)9 (b) 3 (c) 7 (d) 6

Options

A. 9
B. 3
C. 7
D. 6

Answer

D

Explanation

(4) Two parabola are equal if the lengths of their latus rectums are equal.
The length of the latus rectum of `y^(2)=lamdax" is " lamda`.
The equation of the second parabola is
`25{(x-3)^(2)+(y+2)^(2)}=(3x-4y-2)^(2)`
`orsqrt((x-3)^(2)+(y+2)^(2))=(|3x-4y-2|)/(sqrt(3^(2)+4^(2)))`
Here, focus is (3,-2) and the equation of the directrix is
`3x-4y-2=0`
`:.` Length of latus rectum `=2xx` Distance between focus and directrix
`=2|(3xx3-4xx(-2)-2)/(sqrt(3^(2)+(-4)^(2)))|=6`
Thus, the two parabola are equal if `lamda=6`.