The area of the region bounded by the ellipse $\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{16}} = 1$ is

• $20\pi$ sq units

• $20{\pi ^2}$ sq units

• $16{\pi ^2}$ sq units

• $25\pi$ sq units

Correct Option (a)

We have, $\frac{{{x^2}}}{{{5^2}}} + \frac{{{y^2}}}{{{4^2}}} = 1$,
which is ellipse with its axis as coordinate axis.

$\frac{{{y^2}}}{{{4^2}}} = 1 - \frac{{{x^2}}}{{{5^2}}}$
$\Rightarrow$ ${y^2} = 16\left( {1 - \frac{{{x^2}}}{{25}}} \right)$
$\Rightarrow$ $y = \frac{4}{5}\sqrt {{3^2} - {x^2}}$

From the figure, area of the shaded region
$A = 4\int_0^5 {\frac{4}{5}} \sqrt {{5^2} - {x^2}} dx$

$= \frac{{16}}{5}\left[ {\frac{x}{2}\sqrt {{5^2} - {x^2}} + \frac{{{5^2}}}{2}{{\sin }^{ - 1}}\frac{x}{5}} \right]_0^5$

$= \frac{{16}}{5}\left[ {0 + \frac{{{5^2}}}{2}{{\sin }^{ - 1}}1 - 0 - 0} \right] = \frac{{16}}{5} \cdot \frac{{25}}{2} \cdot \frac{\pi }{2} = 20\pi {\rm{sq}}.$ units