Find the required point be `P(x_1, y_1)dot` The tangent to the curve `sqrt(x)+sqrt(y)=4` at which tangent is equally inclined to the axes.

Explanation

We have, `sqrt(x) + sqrt(y)=4`
`rArr x^(1//2)+y^(1//2)=4`
`rArr 1/2.1/(x^(1//2))+1/2.(1/y^(1//2)).(dy)/(dx)=0`
`(dy)/(dx) = -1/2.x^(-1//2)2.y^(1//2)`
`=-sqrt(y/x)`
Since, tangent is equally inclined to the axes.
`therefore (dy)/(dx) = +-1`
`rArr y/x = 1 rArr y=x`
From Eq. (i), `sqrt(y)+sqrt(y)=4`
`rArr 2sqrt(y)=4`
`therefore y=4` and x=4
When y=4, then x=4
So, the required conditions are (4,4).