Find a point on the axis of a parabola

[FedEx]

Homework Statement

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 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 $$(at_{1}^2,2at_{1}) and (at_{2}^2,2at_{2})$$

So the equation of the chord would be $$y(t_{1} + t_{2}) = 2x + 2at_{1}t_{2}$$

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

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.

 

[HallsofIvy]

The theorem says that there exists such a point K. What are the conditions on K that will make $$\frac{1}{PK^2} + \frac{1}{QK^2}$$ independent of t₁ and t₂?

 

[FedEx]

The theorem says that there exists such a point K. What are the conditions on K that will make $$\frac{1}{PK^2} + \frac{1}{QK^2}$$ independent of t₁ and t₂?

Yes they are saying that there exists such a point that $$\frac{1}{PK^2} + \frac{1}{QK^2}$$ is same for all positions of that point and we have to prove this.

 

[HallsofIvy]

My point was that you said:

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.

without any conditions on K. You are not asked to show that but rather find the single point K for which that is true.

 

[FedEx]

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 let's forget about that for a minute, i have shown that the coordinates of the point k would be $$(-at_{1}t_{2},0)$$.

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

$$y = -t_{1}x + 2at_{1} + at_{1}^3$$ 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.