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Homework Statement
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
Homework Equations
We can apply the parametric equations of a parabola.
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.
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