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. $\frac{31}{10}$ sq. units
• B. $\frac{29}{10}$ sq. units
• C. $\frac{21}{10}$ sq. units
• D. $\frac{27}{10}$ sq. units

Answer 1

The correct option is D $\frac{27}{10}$ sq. units

Here, $A\equiv (3,0),B\equiv (0,1)$
Equation of line $AM$ is
$y-1=\frac{0-1}{3-0}(x-0)$
$\Rightarrow x+3y-3=0$

Perpendicular distance of origin from the line is $ON:$

$\Rightarrow \left|\frac{-3}{\sqrt{1^2+3^2}}\right|=\frac{3}{\sqrt{10}}$
Length of $AM=2\left(AN\right)$
$(\because △MNO\cong △ANO)$
$\Rightarrow AM=2\times \sqrt{(OA{)}^2-(ON{)}^2}$
$\Rightarrow AM=2\times \sqrt{9-\frac{9}{10}}=\frac{18}{\sqrt{10}}$
Hence, area of $△AMO=\frac{1}{2}\times AM\times ON$
$=\frac{1}{2}\times \frac{18}{\sqrt{10}}\times \frac{3}{\sqrt{10}}$
$=\frac{27}{10}$ sq. units