Question

P is a point on the ellipse x²/a² + y²/b² = 1 and Q is the corresponding point on the auxiliary circle x² + y² = a² .If the tangent at P meets the major axis at T, then show that QT touches the auxiliary circle.

Answer 1

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.