Consider an ellipse, whose center is at the origin and its major axis is along the x -axis. If its eccentricity is 3 5 and the distance between its foci is 6 , then the area (in sq. units) of the quad

Question

Consider an ellipse, whose center is at the origin and its major axis is along the x -axis. If its eccentricity is 3 5 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:

TestHub · Accepted Answer

The standard equation of an ellipse with its center at the origin and major axis along the $x$ -axis is $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ .

Given eccentricity $e = \frac{3}{5}$ .

The distance between the foci is $2 a e = 6$ .

Substituting $e = \frac{3}{5}$ , we get $2 a (\frac{3}{5}) = 6 \Rightarrow \frac{6 a}{5} = 6 \Rightarrow a = 5$ .

We know that $b^{2} = a^{2} (1 - e^{2})$ .

$b^{2} = 5^{2} (1 - (\frac{3}{5})^{2}) = 25 (1 - \frac{9}{25}) = 25 (\frac{25 - 9}{25}) = 25 (\frac{16}{25}) = 16$ .

So, $b = 4$ .

The vertices of the ellipse are $(\pm a, 0)$ and $(0, \pm b)$ .

These are $(5, 0)$ , $(- 5, 0)$ , $(0, 4)$ , and $(0, - 4)$ .

The quadrilateral formed by these vertices is a rhombus.

The area of a rhombus is $\frac{1}{2} d_{1} d_{2}$ , where $d_{1}$ and $d_{2}$ are the lengths of the diagonals.

Here, $d_{1} = 2 a = 2 (5) = 10$ (along the $x$ -axis) and $d_{2} = 2 b = 2 (4) = 8$ (along the $y$ -axis).

Area $= \frac{1}{2} (2 a) (2 b) = 2 ab = 2 (5) (4) = 40$ sq. units.

The final answer is $40$ .

Mathematics - Ellipse Question with Solution | TestHub

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