Question

The line passing through the extremity A of the major axis and extremity B of the minor axis of the ellipse $x^2+9y^2=9$, meets its auxiliary circle at the point M. Then the area of the triangle with vertices at A, M and the origin ‘O’ is

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

Equation of given ellipse is $\frac{x^2}{9}+\frac{y^2}{1}=1$
Equation of auxiliary circle is $x^2+y^2=\mathrm{9........}\left(1\right)$
Equation of line AB is $\frac{x}{3}+\frac{y}{1}=1\Rightarrow x=3(1-y)$

Putting this in (1), we get $9(1-y{)}^2+y^2=9\Rightarrow 10y^2-18y=0\Rightarrow y=0,\frac{9}{5}$
Thus, y coordinate of ‘M’ is $\frac{9}{5}$
$\Delta OAM=\left(\frac{1}{2}\right)\left(OA\right)\left(MN\right)=\frac{1}{2}\left(3\right)\frac{9}{5}=\frac{27}{10}$