Question

Equation of the curve passing through the point $\left(1 , 2\right)$ such that the intercept on the $x$ -axis cut off between the tangent and origin is twice the abscissa is given by $xy=k$ then find $k$ .

Answer 1

equation of tangent

$\because x-y\frac{d x}{d y}=2x$
$\displaystyle \int \frac{d y}{y}=-\displaystyle \int \frac{d x}{x}$
$\Rightarrow ln\left(y\right)+ln\left(x\right)=ln\left(c\right)$
$\Rightarrow c=2.$
$\Rightarrow xy=2.$