P(2ap, ap^2) and Q(2aq,aq^2) are 2 points on the parabola x^2 = 4ay. Tangents to the parbaola at P and Q intersect at the point T.
a) Show that the equation of the tangent at P is y=px-ap^2.
b) Find the coordinates of T.
c) P and Q move on the parabola so that the line PQ passes through the point (2a,-a). Show that p + q + 1 = pq.
d) Hence, by finding the Cartesian equation of the locus T, show that T lies on a straight line.
e) With the aid of a diagram, carefully explain why the locus of T is not all of the straight line.
Could someone please show me how to do (e). I've found out that the locus of T is x-y+a = 0 and drawn myself a diagram with all the info provided and obtained but I have no clue how to use these to prove (e).
It is meaningless to talk about the point of intersection of two tangents if the tangents are drawn at one and the same point. In the algebra, the point of intersection of the tangents at P and Q is found by solving the equation:
We can divide both sides by to get the solution:
only if ; otherwise we should be dividing by zero - and that's not allowed!