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.

  • 1
    $\begingroup$ can you use $\LaTeX$ please? $\endgroup$ Dec 29, 2015 at 12:31
  • $\begingroup$ @Dr.SonnhardGraubner please help $\endgroup$
    – user101522
    Dec 29, 2015 at 13:14

1 Answer 1


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.

  • $\begingroup$ Can you elaboate it please . $\endgroup$
    – user101522
    Dec 30, 2015 at 12:15
  • $\begingroup$ Can you tell me some other way please!!!!!! $\endgroup$
    – user101522
    Dec 30, 2015 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, 2015 at 15:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.