Question

Area of the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ is

• A. $\pi a b$sq. unit
• B. $\frac { 1 } { 2 } \pi a b$sq. unit
• C. $\frac { 1 } { 4 } \pi a b$sq. unit
• D. None of these

Answer 1

Answer: (A)

$\pi a b$sq. unit

Answer 2

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

$= 4 a b \int _ { 0 } ^ { \pi / 2 } \left( \frac { 1 + \cos 2 \theta } { 2 } \right) d \theta$, {Putting $x = a \sin \theta$}

$= 2 a b \left( \int _ { 0 } ^ { \pi / 2 } d \theta + \int _ { 0 } ^ { \pi / 2 } \cos 2 \theta d \theta \right)$

$= [ \theta ] _ { 0 } ^ { \pi / 2 } + \left[ \frac { \sin 2 \theta } { 2 } \right] _ { 0 } ^ { \pi / 2 } = \pi a b$sq. unit.