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Q.

The line passing through the extremity $A$ of the major axis and extremity $B$ of the minor axis of the
ellipse $x^{2} + 9 y^{2} = 9$ meets the auxiliary circle at the point $M$ . The area of the triangle $A M O$ is

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a

$\frac{31}{10}$

b

$\frac{29}{10}$

c

$\frac{21}{10}$

d

$\frac{27}{10}$

answer is D.

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Detailed Solution

$\frac{x^{2}}{9} + \frac{y^{2}}{1} = 1 \Rightarrow A \equiv (3, 0), B \equiv (0, 1)$

The line $A B : \frac{x}{3} + \frac{y}{1} = 1$ meets the circle $x^{2} + y^{2} = 9$ at $M$

$\begin{array}{l} ∴ x^{2} + {(1 - \frac{x}{3})}^{2} = 9 \Rightarrow \frac{10}{9} x^{2} - \frac{2 x}{3} - 8 = 0 \\ \Rightarrow 5 x^{2} - 3 x - 36 = 0 \Rightarrow x = - \frac{12}{5}, y = \frac{9}{5} \\ ∴ M \equiv (- \frac{12}{5}, \frac{9}{5}) \end{array}$ .

Area of $Δ A M O = \frac{1}{2} |\begin{array}{ccc} 3 & 0 & 1 \\ - \frac{12}{5} & \frac{9}{5} & 1 \\ 0 & 0 & 1 \end{array}| = \frac{27}{10}$

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