Find all curves in the first quadrant such that for any point, the segment of the tangent going from the y intercept to the x intercept is bisected by the point of tangency.

Here's my thoughts ...

Let $\displaystyle (x_0,y_0)$ be the point of tangency.

The tangent has the equation $\displaystyle y=\left.\frac{dy}{dx}\right\vert_{x_0}(x-x_0)+y_0$

The condition that the segment of the tangent going from the y-intercept to the x-intercept be bisected at the point of tangency means:

$\displaystyle y(0)=-\left.\frac{dy}{dx}\right\vert_{x_0}x_0+y_0=2y_0 \leftrightarrow y_0=-\left.\frac{dy}{dx}\right\vert_{x_0}x_0$

... or putting $\displaystyle y(2x_0)=0$ gives the same.

Now I'm stuck.

Thanks for any help!