Prove that there always exists a line and a point for the given curve satisfying the given condition.

Asked Jan 04 '23 at 15:21

Active Jan 04 '23 at 16:05

Viewed 43 times

Let a there be a second degree curve represented by, $$S_1:ax²+2hxy+by²+2gx+2fy+c=0$$ Where atleast one of $a,b,h≠0$ and $a,b,c,h,g,f∈ℝ$

Prove that there always exists a point $S$ and line $L$, which which satisfies, $$\frac{SP}{PM}=e$$

Where $P$ is any point on $S_1$ and $M$ is the foot of perpendicular from $P$ to $L$. And $e\geq0$ is a constant real number.

My approach :

I firstly tried making cases that if, $\Delta=0$ or $≠0$ for curve. So, for $\Delta=0$ curve represents a straight line as it can be factored in to linear expressions. But I can't figure it out for $\Delta≠0$. Can anyone help me out?

edited Jan 04 '23 at 16:05

asked Jan 04 '23 at 15:21

1

Please share your thoughts , your works upon the same question . – Arthur Jan 04 '23 at 15:26
You'd better choose another name for point $S$, it's the same as the curve. – Intelligenti pauca Jan 04 '23 at 16:02
You know that conic sections have focus and directrix, don't you? – Intelligenti pauca Jan 04 '23 at 16:06
By the way: that property is false for a circle, unless you allow $L$ to be at infinity. – Intelligenti pauca Jan 04 '23 at 16:10
Yes $L$ doesn't have any limits, It could be imaginary (in case of imaginary lines on solving $S_1$), and ofcourse can be at infinity. – Jan 04 '23 at 16:15
Yes, I know conics have directrix and focus but I haven't seen any proof regarding this in general. – Jan 04 '23 at 16:16
https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Conic_Section – Intelligenti pauca Jan 04 '23 at 17:54
https://math.stackexchange.com/questions/1833973/prove-that-the-directrix-focus-and-focus-focus-definitions-are-equivalent – Intelligenti pauca Jan 04 '23 at 17:55

Prove that there always exists a line and a point for the given curve satisfying the given condition.

0 Answers0