KeypadSDM, you are not right in assuming that the t and 2t are the x co-ordinates of P and Q, they are the parameters that describe P and Q. Thus, the (admittadely) unclear question means:
P(2at, at^2) and Q(4at, 4at^2) are two points on the parabola x^2 = 4ay. The tangents at P and Q meet at T. Show that the locus of T is 2x^2 = 9ay.
Solution Method: Start with a diagram. P and Q must lie in the sam quadrant, and Q is twice as far from the y-axis as P. Draw tangents at P and Q. They meet at T. We need to find the co-ordinates of T in terms of the parameter T, and then eliminate t to get an equation in x and y for the locus of T.
If x^2 = 4ay, then it is easy to show that y' = x / 2a
At P, m(tang) = t and the equation of PT is y = tx - at^2
At Q, m(tang) = 2t and the equation of QT is y = 2tx - 4at^2
Solving these simultaneously, we get that the co-ordinates of T are (3at, 2at^2).
To find the locus of T, we need to eliminate t from the equations x = 3at, y = 2at^2
Rearranging the first, we get t = x / 3a, and substituting we get y = 2a(x / 3a)^2 = 2ax^2 / 9a^2.
Thus, 9ay = 2x^2, as required.
