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. 1
B. 2
C. 4
D. 6

Answer

D

Explanation

Let us recall that two parabolas are equal if the length of their latusrecta are equal.
Length of the latusrectum of `y^(2)=lamda"x is"lamda`
The equation of the second parabola is
`25|(x-3)^(2)+(y+2)^(2)|=(3x-4y-2)^(2)`
`rArr" "sqrt((x-3)^(2)+(y+2)^(2))=(|3x-4y+2|)/sqrt(3^(2)+(-4))^(2)`
Clearly, it represents a parabola having focus at (3, -2) and equation of the directrix as `3x-4y-2=0`.
`:.` Length of the latusrectum = 2(Distance between focus and directrix)
`=2|(3xx3-4xx-2-2)/sqrt(3^(2)+(-4)^(2))|=6`
Thus, the two parabolas are equal to `lamda=6`.