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

.

So

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.