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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.

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    $\begingroup$ 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? $\endgroup$
    – Darth Geek
    Feb 27, 2023 at 2:38
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    $\begingroup$ "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 $(a,b)$ $\endgroup$
    – Henry
    Feb 27, 2023 at 2:39
  • $\begingroup$ @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 $(a,b)$ $\endgroup$
    – PRD
    Feb 27, 2023 at 2:45
  • $\begingroup$ @DarthGeek, Thanks for your feedback. I stand corrected. I edited my text based on your feedback. $\endgroup$
    – PRD
    Feb 27, 2023 at 2:46

1 Answer 1

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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.

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  • $\begingroup$ 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. $\endgroup$
    – PRD
    Feb 27, 2023 at 4:13
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    $\begingroup$ @PRD All lines are curves, really. $\endgroup$ Feb 27, 2023 at 4:14
  • $\begingroup$ Thanks, I got it. A line can be thought of as a special case of a curve with zero curvature. $\endgroup$
    – PRD
    Feb 27, 2023 at 4:16
  • $\begingroup$ @PRD Please accept my answer if you found it useful. $\endgroup$ Feb 27, 2023 at 4:53

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