Finding curves whose tangents intersect with the x-axis at $(\frac{x}{2},0)$

Question

I have to find the family of curves in $\mathbb{R}^2$ with this property:

The tangent in an arbitrary point on the curve does intersect with the x-axis in $(\frac{x}{2}, 0)$.

I think I have to make a differential equation first, but how do I do that?

Answer 1

Let $P=(\bar x,f(\bar x))$ be a point of the curve. The tangent at this point has equation: $$ y-f(\bar x)=f'(\bar x)(x-\bar x) $$

We want that the point $(\bar x/2,0)$ is on this line, so we have:

$$ -f(\bar x)=f'(\bar x)(\frac{\bar x}{2}-\bar x) $$

since this have to be true for all $\bar x$ we have the equation:

$$ f(x)=\frac{x}{2}f'(x) $$ that has simple solution.