Hi guys!

I am baffled by my own solution to the following parabola question. Having got the right answer, I find that I don't like it! Can anyone help?

The parabola is and at point the tangent is found to be . We have to find the coordinates of the point of intersection T of the tangent at P and the tangent at .

In the derivation of the tangent at P the gradient is found to be 1/p.

To get the tangent at Q, given that P and Q have the same coordinate , I would expect the intersection to be on the x-axis (since is symmetrical). The answer does not confirm this, so that's my first question - why not?

If I put the coords of Q into the parabola I get:

I therefore take the gradient at Q to be -1/p, then the equation of the tangent follows:

call this equation 1

The tangent at P is

subtracting gives

which is the correct answer.

The thing that puzzles me is that to get the answer I am assuming the gradient at Q is symmetrical i.e. -1/p (since the gradient at P is 1/p). Also if I express this gradient as 1/q which is the same thing and substitute into equation 1, I get:

and equation 1 is:

subtracting:

A different result! All these ps and qs are swimming before my eyes. Have I made a slip somewhere?