1994 KINGS Rectangular Hyperbola + parabola (1 Viewer)

[asianese]

b) A tangent to the parabola meets the hyperbola in the points P, Q.
i) Show that the equation of the tangent at on the parabola is
ii) Show that the x coordinates of P and Q are given by the equation
iii) Deduce the cartesian equation of the locus of the midpoint M of the interval PQ.

I've done ii) and I got that equation. Stuck on iii)

I noticed that the equation is a quadratic in x, so that the sum of roots divided by 2 is the midpoint, so . Now PQ is on the tangent, so And there's where I'm stuck...don't know how to get rid of the y1..

Any help is appreciated!

 

[Carrotsticks]

When subbing in the midpoint into the eqn of PQ, why did you sub in -ax_1/2 as opposed to just -x_1/2 ?

 

[asianese]

Oh wow that's a good point. lolol DERP. thanks!