Question

A 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 auxillary circle at the point $M$. Then the area of the triangle with vertices at $A,M$ and the origin $O$ is

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

Answer 1

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

$\frac{x^2}{a}+y^2=1$

Equation of line AB

$\frac{x}{3}+\frac{y}{1}=1$

$x+3y=3$

$(3\cos \theta ,3\sin \theta )$

$3\cos \theta +9\sin \theta =3$

$\cos \theta +3\sin \theta =1$

$\cos \theta =1-3\sin \theta$

${\cos }^2\theta =1+9{\sin }^2\theta -6\sin \theta$

$1-{\sin }^2\theta =1+9{\sin }^2\theta -6\sin \theta$

$10{\sin }^2\theta -6\sin \theta =0$

$\sin \theta =0$, $\sin \theta =\frac{6}{10}$

Area$=\frac{1}{2}\left|\begin{array}{cccc}0& 3\cos \theta & 3& 0\\ 0& 3\sin \theta & 0& 0\end{array}\right|$

$=\frac{1}{2}9\sin \theta$

$=\frac{1}{2}9\frac{6}{10}$

$=\frac{27}{10}$ sq. units.