Question

Find area of ellipse $25x^2+4y^2=100.$

Answer 1

Method I - Directly apply the formula

Area $=\pi ab$

$=\pi \times 2\times 5$

$=10\pi squnits.$

Method II- Complete derivative of $\pi ab$.

$\frac{y^2}{25}=1-\frac{x^2}{4}$

$\therefore y=5\sqrt{1-\frac{x^2}{4}}$ ( in I quadrant)

$\therefore \frac{1}{4}\times$ Area of ellipse $={\int }_0^{2}5\sqrt{1-\frac{x^2}{4}}dx$

Substituting $x=a\sin t=2\sin t\Rightarrow dx=2\cos tdt$

Area of ellipse $={\int }_0^{\pi /2}10{\cos }^{2}tdt$

changing the limits appropriately.

$\therefore$ Area $=10\times {\int }_0^{\pi /2}10{\cos }^{2}tdt$

$=10\pi sq.units$.