The equation of the curve in which the portion of the tangent included between the coordinate axes is bisected at the point of contact, is

a hyperbola

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

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

This cuts the coordinate axes at $A\left(x-y \frac{d x}{d y}, 0\right)$ and $B\left(0, y-x \frac{d y}{d x}\right)$

It is given that $P(x, y)$ is the mid-point of $A B$.

∴  $x-y \frac{d x}{d y}=2 x$ and $y-x \frac{d y}{d x}=2 y$

$\Rightarrow x+y \frac{d x}{d y}=0$ and $y+x \frac{d y}{d x}=0$

$\Rightarrow x d y+y d x=0$ and $y d x+x d y=0$

$\Rightarrow d(x y)=0 \Rightarrow x y=C$

Clearly, it represents a rectangular hyperbola.