Question

Assertion :The area of the ellipse $\frac{x^2}{3}+\frac{y^2}{2}=1$ is less than the area bounded by $\frac{\left|x\right|}{3}+\frac{\left|y\right|}{2}\le 1$. Reason: The length of the major axis of the ellipse $\frac{x^2}{3}+\frac{y^2}{2}=1$ is more than the distance between the points of $\frac{\left|x\right|}{3}+\frac{\left|y\right|}{2}=1$ on the axis.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

$\frac{x^2}{{\left(\sqrt{3}\right)}^2}+\frac{y^2}{{\left(\sqrt{2}\right)}^2}=1$

Area of ellipse $=\pi \left(\sqrt{3}\right)\left(\sqrt{2}\right)=3.14\times 1.7\times 1.41=7.8$ sq. units.

Area bounded by $\frac{\left|x\right|}{3}+\frac{\left|y\right|}{2}=1$ or $2\left|x\right|+3\left|y\right|=6$ is the area of the parallelogram as shown.

$=4\times \frac{1}{2}\times 3\times 2=12$ sq. units

Also, length of the major axis $=2\sqrt{6}<3+3.$