the eqn of line joing 2 pointswhose co-ordinates are (acosr, bsinr) and (acost, bsint)
x/acos(r+t)/2 + y/bsin(r+t)/2 = cos(r-t/2)
now put the values of r and t as α-β and α+β
This will be the chord of contact of the tangets from the ext point whose co-ordinates need to found.
write eqn of chord of contact at (x1y1) and compare with above eqn to get value of (x1y1)