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JEE Main

Exam: JEE Main
Subject: Mathematics
Chapter: Ellipse

JEE Main · Mathematics · Ellipse

The line passing through the extremity A of the major axis and...

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

The line passing through the extremity A of the major axis and extremity B of the minor axis of the ellipse x^ 2 + 9y^

Options

A $\dfrac{31}{10}$
B $\dfrac{29}{10}$
C $\dfrac{21}{10}$
D $\dfrac{27}{10}$

Correct answer

D. $\dfrac{27}{10}$

Step-by-step solution

Equation of auxiliary circle is

$x^{2} + y^{2} = 9 . . \left(i\right)$

The line passing through the extremity A of the major axis and extremity B of the minor axis of the ellipse x^ 2 + 9y^

$∴$ Equation of $AM$ is \dfrac{x}{3} + \dfrac{y}{1} = $1 . . . \left(ii\right)$

On solving Eqs. (i) and (ii), we get

$M \left(- \dfrac{12}{5}, \dfrac{9}{5}\right)</p><p>Now, area of \Delta$ AOM = $\dfrac{1}{2}$ OA \timesMN = $\dfrac{27}{10}sq units</p><p>Hence, option (d) is correct.</p>$

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