Equation of the curve passing through the point$(1,\ 2)$

such that the intercept on the

$x-$axis cut off between the tangent and origin is twice the abscissa is given by:

The equation of the tangent at any point $p(x, y)$ is

$Y-y=\frac{d y}{d x}(x-x)$

Given that length of intercept on $x$-axis $=2(x$-co-ordinates of $p)$

$\left|x-y \frac{d x}{d y}\right|=2 x$

$\Rightarrow \frac{d y}{d x}=\frac{-y}{x} \text { or } \frac{d y}{d x}=\frac{y}{3 x}$

On integrating we get,

$x y=c \text { or } 3 \ln |y|=\ln |x|+\ln |k|$

Since, the curve passes through $(1,2)$

$\begin{array}{l}\Rightarrow \quad x y=c \\ \Rightarrow \quad 1 \times 2=c \\ \therefore \quad 0=2\end{array}$

$\therefore$ So, the equation will be,

$x y=2$