1. Tardigrade
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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 + 9y2 = 9 meets its auxiliary circle a t the point M. Then, the area (insqunits) of the triangle with vertices at A, M and the origin O 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 $x^2 + 9y^2 = 9$ meets its auxiliary circle a t the point M. Then, the area (insqunits) of the triangle with vertices at A, M and the origin O is

IIT JEEIIT JEE 2009Conic Sections

Solution:

Equation of auxiliary circle is
$x^{2}+y^{2}=9 ....$(i)
Equation of $A M$ is $ \frac{x}{3}+\frac{y}{1}=1.....$(ii)
image
On solving Eqs. (i) and (ii), we get $M\left(-\frac{12}{5}, \frac{9}{5}\right)$.
Now, area of $\triangle A O M=\frac{1}{2} \cdot O A \times M N=\frac{27}{10}$ sq units