Question

Assertion :The area of the region $3x^2+4y^2=12$ will be greater than the area bounded by $3\left|x\right|+4\left|y\right|\le 12$ Reason: The length of major axis of the ellipse $3x^2+4y^2=12$ is less than the distance between the points of $3\left|x\right|+4\left|y\right|\le 12$ on x-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 incorrect but Reason is correct
• D. Both Assertion and Reason are incorrect

Answer 1

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

Area bounded by the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is $\pi ab$

$\therefore$ Area bounded by the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$ is

$\pi \times 2\sqrt{3}=3.14\times 2\left(1.732\right)=6.28\times 1.732=10.87696$
i.e. $10.88$ square units (Approximately)

Again area bounded by $3\left|x\right|+4\left|y\right|\le 12$
$=4$ Area of $△AOD$
$=2\times 3\times 4$
$=24$ square units which is greater than the area bounded by the ellipse

Hence Assertion (A) is false.
Length of major axis $\left(LM\right)$ $=2\times 2=4$
Also the distance between the points $(4,0)$ & $(-4,0)$ is $8$

$\therefore$ Length of major axis of ellipse < the distance between the points of $3\left|x\right|+4\left|y\right|\le 12$ on x-axis so Reaon (R) is correct

Assertion (A) is false & Reason (R) is true.