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Prove that on the axis of any parabola $y^2=4ax$ there is a certain point $K$ which has the property that,if a chord $PQ$ of the parabola be drawn through it ,then $$\frac{1}{PK^2}+\frac{1}{QK^2}$$ is same for all positions of the chord.Find also the coordinates of the point $K$.

We can apply the parametric equations of a parabola Let the points $P$ and $Q$ be $(at_1^{2},2at_1)$ and $(at_2^{2}, 2at_2)$

So the equation of the chord would be $$y(t_1+t_2)=2x+2at_1t_2$$

Hence from there we have that the coordinates of $K$ are $(−at_1t_2,0)$

Now our aim is to show that $\frac{1}{PK^2}+\frac{1}{QK^2}$ is independent of $t_1$ and $t_2$. I tried and applied the distance formula but no benefit.

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    $\begingroup$ can you use $\LaTeX$ please? $\endgroup$ – Dr. Sonnhard Graubner Dec 29 '15 at 12:31
  • $\begingroup$ @Dr.SonnhardGraubner please help $\endgroup$ – user101522 Dec 29 '15 at 13:14
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Hint: This is beyond easy ! :-$)$ The sum in question stays constant, regardless of the position of P, correct ? So just let $K=(b,0),$ and then take two positions for P: when P is right above $($or right below$)$ K, and when $P\to\infty$. Where is Q in both these cases ? Can you deduce the value of b from equating the two sums ? I just did, and used GeoGebra to verify the result.

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  • $\begingroup$ Can you elaboate it please . $\endgroup$ – user101522 Dec 30 '15 at 12:15
  • $\begingroup$ Can you tell me some other way please!!!!!! $\endgroup$ – user101522 Dec 30 '15 at 13:47
  • $\begingroup$ @user101522: Were you at the very least able to determine the position of Q $($and the lengths of the two segments$)$ in each of the two cases I mentioned ? $\endgroup$ – Lucian Dec 30 '15 at 15:02

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