The area of the region bounded by the ellipse | vedclass

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Question

(C) The area of an ellipse given by $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is given by the formula $A = \pi ab$. Given that the area is $\frac{\pi}{6

Vedclass · Accepted Answer

$4x^2 + 9y^2 = 1$

Vedclass · Answer

(C) The area of an ellipse given by $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is given by the formula $A = \pi ab$. Given that the area is $\frac{\pi}{6}$,we have $\pi ab = \frac{\pi}{6}$,which implies $ab = \frac{1}{6}$. Let us check the options: For option $C$,the equation is $4x^2 + 9y^2 = 1$,which can be written as $\frac{x^2}{1/4} + \frac{y^2}{1/9} = 1$. Here,$a^2 = \frac{1}{4} \implies a = \frac{1}{2}$ and $b^2 = \frac{1}{9} \implies b = \frac{1}{3}$. The area is $\pi ab = \pi 	imes \frac{1}{2} 	imes \frac{1}{3} = \frac{\pi}{6}$. Thus,option $C$ is correct.

The area of the region bounded by the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is $\frac{\pi}{6}$ sq. units. Which of the following is a possible equation of the ellipse?

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