P = (a cos θ, b sin θ) and Q ] (a cos θ, a sin θ). The tangent at P to the ellipse is
xcosθ/a + ysinθ/b = 1
Therefore, T = (a sec θ, 0) and the equation of the line TQ is
= -x cotθ + acosec θ
This line touches the auxiliary circle x2 + y2 = a2. This implies
a2 cosec2 θ = a2 (1 + cot2θ) = a2 cosec2θ
which is true. Hence, TQ touches the auxiliary circle.