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

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.

• Note that $y = x + C$ does not pass through the origin unless $C = 0$. Are you sure the differential equation does not yield $y = Cx$ instead? Feb 27, 2023 at 2:38
• "Among the curves whose tangent always passes through the origin, find the one that passes through point $(a,b)$" is difficult for me to understand. What curves? All of its tangents? Or does it only have one tangent? You came to the conclusion you are taking about $y=Cx$ (so not curvy at all) and the next step would be to say $C=\frac ba$ so it passes through $(a,b)$ Feb 27, 2023 at 2:39
• @Henry. Thank you for trying to understand the question. Based on my professor, among the curves whose all tangents pass through the origin, I was asked to find the one that passes through point $(2,4)$. But I would like to know what the general equation of will be the any of that curve using the arbitrary point $(a,b)$
– PRD
Feb 27, 2023 at 2:45
• @DarthGeek, Thanks for your feedback. I stand corrected. I edited my text based on your feedback.
– PRD
Feb 27, 2023 at 2:46

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.

• Thank you. I think my problem is that I assume that the curve should be a curve line that is why I am hesitant with my answer. So, a line could be considered as a curve.
– PRD
Feb 27, 2023 at 4:13
• @PRD All lines are curves, really. Feb 27, 2023 at 4:14
• Thanks, I got it. A line can be thought of as a special case of a curve with zero curvature.
– PRD
Feb 27, 2023 at 4:16
• @PRD Please accept my answer if you found it useful. Feb 27, 2023 at 4:53