Question

The abscissa of the point on the curve $\sqrt{xy}=a+x,$ the tangent at which cuts off equal intercepts from the coordinate axes is

Answer 1

$xy=x^2+a^2+2ax$
$\Rightarrow y=\frac{a^2}{x}+2a+x$
$\frac{dy}{dx}=1-\frac{a^2}{x^2}$
Since tangent cuts off equal intercepts from the axes,
$y^{\prime }=-1$
$1-\frac{a^2}{x^2}=-1$
$2=\frac{a^2}{x^2}$ or $2x^2=a^2$ or $x=\pm \frac{a}{\sqrt{2}}$