Question

Let
$A1 = {(x,y):|x| <= y^2,|x|+2y≤8}$
and
$A2 = {(x,y) : |x| +|y|≤k}.$
If 27(Area A1) = 5(Area A2), then k is equal to :

Answer 1

The correct answer is 6

Fig.

Required area (above x -axis)
$A_1 = 2 \int_{4}^{8} (8 - \frac{x}{2} - \sqrt{x}) \, dx$
$= 2 \left(16 - \frac{16}{4} - \frac{8}{3/2}\right) = \frac{40}{3}$
and
$A_2 = 4 \left(\frac{1}{2}. k^2\right) = 2k^2$

Fig.

$\therefore 27 \cdot \frac{40}{3} = 5 \cdot (2k^2)$
⇒ k = 6

*A1

Fig.

Which tends to infinity if not mentioned above x -axis