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