You have found the co-ordinates of R and T.
Find the mid point R and T. That will be the co-ordinates of Q.
P is a point on an ellipse whose major axis is AB the tangent at P meets the minor axis at Q. PA and PB cut the minor axis at R and T. Prove that Q bisects RT.
I put A=(a,0) and B=(-a,0)
equation of tangent at P
Point where tangent cuts minor axis, when x=o
Now I don't know what to do. I can't show that RT=2RQ