A) \[\pi \,ab\]sq. unit
B) \[\frac{1}{2}\pi \,ab\]sq. unit
C) \[\frac{1}{4}\pi \,ab\]sq. unit
D) None of these
Correct Answer: A
Solution :
Since the given equation contains only even powers of x and only even powers of y, the curve is symmetrical about y-axis as well as x-axis. \ Whole area of given ellipse \[=4(\text{area }\,\text{of}\,BCO)=4\times \int_{0}^{a}{y\,dx=4\int_{0}^{a}{\frac{b}{a}\sqrt{{{a}^{2}}-{{x}^{2}}}}dx}\] \[=4ab\int_{0}^{\pi /2}{\left( \frac{1+\cos 2\theta }{2} \right)\,d\theta }\], {Putting \[x=a\sin \theta \]} \[=2ab\left( \int_{0}^{\pi /2}{\,\,d\theta +\int_{0}^{\pi /2}{\,\,\,\cos 2\theta \,d\theta }} \right)\] \[=[\theta ]_{0}^{\pi /2}+\left[ \frac{\sin 2\theta }{2} \right]_{0}^{\pi /2}=\pi ab\]sq. unit.You need to login to perform this action.
You will be redirected in
3 sec