Practicing Success
If $x \sqrt{1+y}+y \sqrt{1+x}=0, x \neq y$ then $2 \frac{d y}{d x}+y$ at x = -3 is equal to |
-1 1 2 -2 |
-2 |
$x \sqrt{1+y}+y \sqrt{1+x}=0$ $\Rightarrow x \sqrt{1+y}=y \sqrt{1+x}$ Squaring both sides $\Rightarrow x^2(1+y)=y^2(1+x)$ so $x^2+x^2 y=y^2+y^2 x$ $\Rightarrow x^2-y^2+x y(x-y)=0$ $(x-y)(x+y+x y)=0$ ⇒ x + y + xy = 0 (as given x ≠ y) at x = -3 -3 + y - 3y = 0 so $y = \frac{-3}{2}$ differentiating wrt x $1+\frac{d y}{d x}+y+\frac{x d y}{d x}=0$ at x = -3 , y = -3/2 $1+\frac{d y}{d x}-\frac{3}{2}-3 \frac{d y}{d x}=0 \Rightarrow \frac{d y}{d x}=\frac{-1}{4}$ $\frac{2 d y}{d x}+y=-\frac{1}{2}-\frac{3}{2}=-2$ Option: D |