Consider an ellipse, whose centre is at the origin and its major axis is along the x-axis.  If its eccentricity is   and the distance between its foci is 6, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is : Option 1) 8 Option 2) 32 Option 3) 80 Option 4) 40

As we learnt in

Length of major axis -

$2a$

- wherein

$a\rightarrow$ Semi major axis

Length of major axis -

$2b$

- wherein

$b\rightarrow$ Semi minor axis

Eccentricity -

$e= \sqrt{1-\frac{b^{2}}{a^{2}}}$

- wherein

For the ellipse

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

$e=\frac{3}{5}$

$2ae=6\; \Rightarrow ae=3$

Hence a=5

$b^{2}=a^{2}(1-e^{2})=>b=4$

Area of quadrilateral ABCD = 4 ar $\Delta AOB$

$=4\times \frac{1}{2}\times a\times b$

$=4\times \frac{1}{2}\times 5\times 4$

=4 $\times$ 10 = 40

Option 1)

8

This option is incorrect

Option 2)

32

This option is incorrect

Option 3)

80

This option is incorrect

Option 4)

40

This option is correct

N

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