# Find a point on the axis of a parabola

1. Jan 29, 2008

### FedEx

1. The problem statement, all variables and given/known data

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

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

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.

Last edited: Jan 29, 2008
2. Jan 29, 2008

### HallsofIvy

Staff Emeritus
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?

3. Jan 29, 2008

### FedEx

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.

4. Jan 29, 2008

### HallsofIvy

Staff Emeritus
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.

5. Jan 29, 2008

### 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 lets 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.

Last edited: Jan 29, 2008