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  4. The line passing through the extremity A of the major axis and extremity B of the minor axis of the ellipse x2+9 y2= 9 meets its auxiliary circle at the point M. Then the area of the triangle with vertices at A, M, and O (the origin) is

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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 its auxiliary circle at the point $M$. Then the area of the triangle with vertices at $A, M$, and $O$ (the origin) is

Conic Sections

Solution:

The equation of line $A M$ is
$x+3 y-3=0$
Perpendicular distance of line from the origin $=\frac{3}{\sqrt{10}}$
image
Length of $A M=2 \sqrt{9-\frac{9}{10}}=2 \times \frac{9}{\sqrt{10}}$
or Area $=\frac{1}{2} \times 2 \times \frac{9}{\sqrt{10}} \times \frac{3}{\sqrt{10}}=\frac{27}{10}$ sq. units.