p is a variable point on the parabola x^2 = -4y. The tangent from P cuts the parabola x^2 = 4y at Q and R. Show that 3x^2 = 4y is the equation of the locus of the midpoint of the chord RQ.
Let P have co-ordinates on the lower parabola.
The slope of the tangent is , and you can see from a sketch that this is equal to
divided by a crucial advance on the x-axis that turns out (as a result) to be
And from this you can deduce that the constant in the tangent-line equation is
is an equation for the tangent line which you can set equal to the upper parabola that it has to (twice) intersect...
So solve that equation for x, to get the x-values (and after that the y's) of the points of intersection in terms of a, and then finally their mean average (mid-point) in terms of a. Hopefully that final y-value will be three-quarters the square of the x.