Question

A tangent at any point to the ellipse 4x² + 9y² = 36 is cut by the tangent at the extremities of the major axis at T and T¢. The circle on TT¢ as diameter passes through the point –

• A. (0, $\sqrt{5}$)
• B. ($\sqrt{5}$, 0)
• C. (2, 1)
• D. (0 , –$\sqrt{5}$)

Answer 1

Answer: (B)

($\sqrt{5}$, 0)

Answer 2

Any point on the ellipse is P(3 cos q, 2 sin q).

Equation of the tangent at P is $\frac{x}{3}$ cos q + $\frac{y}{2}$sin q = 1

Which meets the tangents x = 3 and x = –3 at the extremities of the major axis at

T $\left( 3,\frac{2(1 - \cos\theta)}{\sin\theta} \right)$ and T¢$\left( –3,\frac{2(1 + \cos\theta)}{\sin\theta} \right)$

Equation of the circle on TT¢ as diameter is

(x – 3) (x + 3) + $\left( y - \frac{2(1 - \cos\theta)}{\sin\theta} \right)$

$\left( y - \frac{2(1 + \cos\theta)}{\sin\theta} \right)$ = 0

Ž x² + y² – $\frac{4}{\sin\theta}y - 5 = 0$, which passes through ($\sqrt{5}$,0)