The sum of the intercepts made on the axes of coordinates by any tangent to the curve ` sqrt(x)+sqrt(y)=sqrt(a)` is equal to

Options

A. `a`
B. `2a`
C. `2sqrt(a)`
D. None of these

Answer

A

Explanation

Given `sqrt(x)+sqrt(y)=sqrt(a)implies1/(2sqrt(x))+1/(2sqrt(y))(dy)/(dx)=0`
`implies(dy)/(dx)=-(sqrt(y))/(sqrt(x))`
Hence tangent at `(x,y)` is
`Y-y=-(sqrt(y))/(sqrt(x))(X-x)`
`impliesX(sqrt(y))+Y(sqrt(x))=sqrt(xy)(sqrt(x)+sqrt(y))=sqrt(axy)`
`impliesX/(sqrt(a)sqrt(x))+y/(sqrt(a)sqrt(y))=1`
Clearly, its intercepts on the axes are `sqrt(a)sqrt(x)` and `sqrt(a)sqrt(y)`.
Sum of intercepts `=sqrt(a)(sqrt(x)+sqrt(y))=sqrt(a).sqrt(a)=a`