Length of the latus rectum of the parabola `sqrt(x) +sqrt(y) = sqrt(a)` is (a) `a sqrt(2)` (b) `(a)/(sqrt(2))` (c) `a` (d) `2a`

Options

A. `a sqrt(2)`
B. `(a)/(sqrt(2))`
C. a
D. 2a

Answer

A

Explanation

`sqrt(x) = sqrt(a) - sqrt(y)`
`rArr x = a +y -2 sqrt(ay)`
`rArr (x-y-a)^(2) = 4ay`
`rArr x^(2) + (y+a)^(2) -2x (a+y) = 4ay`
`rArr x^(2) + y^(2) - 2xy +2ay +a^(2) -2ax =4ay`
`rArr x^(2) + y^(2) -2xy =2ax +2ay -a^(2)`
`rArr (x-y)^(2) =2a (x+y-(a)/(2))`
Axis is `x -y =0`
`((x-y)/(sqrt(2)))^(2) =(2a)/(2) ((x+y-(a)/(2))/(sqrt(2))) xx sqrt(2)`
`rArr ((x-y)/(sqrt(2)))^(2) =sqrt(2)a ((x+y-(a)/(2))/(sqrt(2)))`
`:.` Length `L.R. = asqrt(2)`