Question

Sol. Put ( $a - rcos\theta,2 - rsin\theta$ ) to eqn. of ellipse.

$\Rightarrow \frac{a^{2} + r^{2}\cos^{2}\theta - 2arcos\theta}{9} + \frac{4 + r^{2}\sin^{2}\theta - 4rsin\theta}{4}$ -1 = 0

$\begin{matrix} \\ \Rightarrow \ (PA)(PD) = \frac{\frac{a^{2}}{9}}{\frac{\cos^{2}\theta}{9} + \frac{\sin^{2}\theta}{4}} \end{matrix}$

$$= \frac{4a^{2}}{4\cos^{2}\theta + 9\sin^{2}\theta}$$

Put ( $a - rcos\theta,2 - rsin\theta$ ) to equation $xy = 0$

$$\begin{matrix} \Rightarrow & 2a + r^{2}sin\theta cos\theta - (asin\theta + 2cos\theta)r = 0 \\ \Rightarrow & \ (PB)(PC) = \frac{2a}{sin\theta cos\theta} \\ & \ PA \cdot PD = PB \cdot PC \\ \Rightarrow & \frac{4a^{2}}{4\cos^{2}\theta + 9\sin^{2}\theta} = \frac{2a}{sin\theta \cdot cos\theta} \\ \Rightarrow & a = \frac{4cot\theta + 9tan\theta}{2} \\ \Rightarrow & \ |a| \geq \sqrt{4 \times 9} = 6 \\ \Rightarrow & a \in ( - \infty, - 6\rbrack \cup \lbrack 6,\infty) \end{matrix}$$