$\text{Each of the four inequalities given below defines a region in the xy plane. One of these four regions does not have }$

$\mathrm{the} \mathrm{following} \mathrm{property}.\text{ For any two points }\left(x_1,y_1\right)\text{ and }$$\left(x_2,y_2\right)\text{ in the region, the point }\left(\left(x_1+x_2\right)/2,\left(y_1+y_2\right)/2\right)\text{ is }$

$\text{ also in the region. The inequality defining this region is }$

detailed solution

Correct option is C

x2+2y2≤1 represents the interior region of a ellipse, where on taking any two points,the midpoint of that segment will also lie inside that circle.Max⁡{|x|,|y|}≤1 or |x|≤1,|y|≤1 or −1≤x≤1 and −1≤y≤1which represents the interior region of a square with its sides x=±1 and y =±1, in which for any two points, their midpoint also lies inside the region.x2−y2≤1, represents the exterior region of a hyperbola in  which we take two points (4,3) and (4,−3) . Then their mid-  point (4,0) does not lie in the same region (as shown in the figure)y2≤x represents the interior region of a parabola in which for  any two points, their midpoint also lies inside the region.