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Find a point on the axis of a parabola

  1. Jan 29, 2008 #1
    1. The problem statement, all variables and given/known data

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


    2. Relevant equations

    We can apply the parametric equations of a parabola.


    3. The attempt at a solution

    Let the points P and Q be [tex](at_{1}^2,2at_{1}) and (at_{2}^2,2at_{2})[/tex]

    So the equation of the chord would be [tex]y(t_{1} + t_{2}) = 2x + 2at_{1}t_{2}[/tex]

    Hence from there we have that the points of K are [tex](-at_{1}t_{2},0)[/tex]

    Now our aim is to show that [tex]\frac{1}{PK^2} + \frac{1}{QK^2}[/tex] is independent of [tex]t_{1} and t_2{}[/tex]. I tried and applied the distance formula but no benefit.
     
    Last edited: Jan 29, 2008
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  3. Jan 29, 2008 #2

    HallsofIvy

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    The theorem says that there exists such a point K. What are the conditions on K that will make [tex]\frac{1}{PK^2} + \frac{1}{QK^2}[/tex] independent of t1 and t2?
     
  4. Jan 29, 2008 #3
    Yes they are saying that there exists such a point that [tex]\frac{1}{PK^2} + \frac{1}{QK^2}[/tex] is same for all positions of that point and we have to prove this.
     
  5. Jan 29, 2008 #4

    HallsofIvy

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    My point was that you said:


    without any conditions on K. You are not asked to show that but rather find the single point K for which that is true.
     
  6. Jan 29, 2008 #5
    Agreed. But we also have to prove that. The question says "Prove that on the axis of any parabola ...........coordinates of the point K.

    And lets forget about that for a minute, i have shown that the coordinates of the point k would be [tex](-at_{1}t_{2},0)[/tex].

    So that is done.

    But when it comes to proving,i am completely at sea.

    But i think we can consider that chord to be a normal at the point P. If we do so we can get the equation of the normal as

    [tex] y = -t_{1}x + 2at_{1} + at_{1}^3 [/tex] and at the same time we can also consider the equation of the tangent passing through P and than we can consider a tangent at Q which will intersect the tangent at P and then we MAY get some relation.
     
    Last edited: Jan 29, 2008
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