Area of the ellipse x 2 a 2 + y 2 b 2 = 1 is:

Question

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

Smart Achievers · Accepted Answer

(1)

Ans. $π a b$ sq. units

Explanation:

Since the given equation contains only even powers of $x$ and only even powers of $y$ , the curve is symmetrical about the $Y$ -axis as well as the $X$ -axis.

$∴$ Whole area of the given ellipse

$= 4 (Area of BCO)$

$= 4 \int_{0}^{a} y d x$

$= 4 \int_{0}^{a} \frac{b}{a} \sqrt{a^{2} - x^{2}} d x$

$= 4 a b \int_{0}^{π / 2} (\frac{1 + \cos 2 θ}{2}) d θ$ [Putting $x = a \sin θ$ ]

$= 2 a b (\int_{0}^{π / 2} d θ + \int_{0}^{π / 2} \cos 2 θ d θ)$

$= 2 a b [{[θ]}_{0}^{π / 2} - {[\frac{\sin 2 θ}{2}]}_{0}^{π / 2}]$

$= 2 a b [\frac{π}{2} - 0 - (0 - 0)]$

$= π a b sq. units.$

Solution

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

1 $π a b$ sq. units

2 $\frac{1}{2} π a b$ sq. units

3 $\frac{1}{4} π a b$ sq. units

4 None of the above

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Solution

Q. Area of the ellipse x2a2+y2b2=1 is:

1 πab sq. units

2 12πab sq. units