First slide

Ellipse

Question

${The line passing through the extremity A of the major axis and extremi ty B of the minor axis of the ellipse x}^{2} {+9y}^{2} =9 meets its$
$auxiliary circle at the point M Then the area of the triangle with vertices at A, M, and O (the origin) is .$

Moderate

A

$\frac{27}{10}$
B

$\frac{31}{10}$
C

$\frac{29}{10}$
D

$\frac{21}{10}$

Solution

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

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