Question

Find all the curves possessing the following property; the segment of the tangent between the point of tangency & the $x$-axis is bisected at the point of intersection with the $y$-axis.

Answer 1

Equation of tangent at $P(x,y)$ is

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

It meets $x$-axis at $Q(x,0)$

$-y=\frac{dy}{dx}(X-x)x-y\frac{dy}{dx}=XQ\equiv (x-y\frac{dy}{dx},0)$

$PQ$ bisected at point of intersection at $y$-axis let $R(0,y)$

$\Rightarrow \frac{x+x-y\frac{dy}{dx}}{2}=02x=y\frac{dy}{dx}\int \frac{dy}{dx}=\int \frac{dx}{2x}\log y=\frac{1}{2}\log x+\log C{y}^2=Cx$