Among the curves whose all tangents pass through the origin, find the one that passes through point $(a,b)$.

Question

Among the curves whose all tangents pass through the origin, find the one that passes through point $(a,b)$.

Here is my solution but my answer seems incorrect.

Let $f(x)$ be the function of the curve. At $(t, f(t))$, the function $y=f(x)$ has a tangent line $y=f'(t)(x-t)+f(t)$.

Since the tangent line passes through the origin, we get

$0=f'(t)(-t)+f(t)$

$tf'(t)=f(t)$,

which can be written as the differential equation $xy'=y$.

After solving the differential equation, I got the family of curve $y = Cx$, which is are composed of straight lines passing through the origin. I am stuck here and don't know what the next step should be. Please feel free to share your thoughts. Thank you in advance.

Answer 1

Now just substitute $x=a,y=b$ into $y=Cx$ and get $C=\frac ba$. If $(a,b)$ is not the origin itself, $y=\frac bax$ is unique.