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

Answer 1

Answer: (C)

$\frac{27}{10}$

The equation of ellipse is $\frac{x^2}{9}+y^2=1$

The length of semi-major axis is $a=3$ and the length of semi-minor axis is $b=1$

The coordinates of point $A$ is $(3,0)$ and the coordinates of point $B$ is $(0,1)$

The equation of line passing through $A,B$ is $x+3y=3$

The equation of an auxiliary circle of an ellipse is $x^2+y^2=9$

The line $AB$ cuts $x^2+y^2=9$ at point $M$

By solving the above equations

The coordinates of point $M$ are $(-\frac{12}{5},\frac{9}{5})$

The area of triangle $AMO$ is $\frac{1}{2}|0(0-\frac{9}{5})+3(\frac{9}{5}-0)-\frac{12}{5}(0-0)|=\frac{27}{10}$