Each of the four inequalities given below defines a region in the xy plane. One of these four regions does not have the following property. For any two points `(x_1,y_1) and (x_2,y_2)` in the region the point `((x_1+x_2)/2*(y_1+y_2)/2)` is also in the region. The inequality defining this region is`(1) x^2 + 2y^2 ≤ 1 (2)Max {|x| , | y| ≤ 1 (3) x^2 – y^2 ≤ 1 (4) y^2 – x ≤ 0`

Explanation

We have given points `A(x_(1),y_(1))` and `B(x_(2),y_(2))` and its midpoint is C`((x_(1)+x_(2))/(2),(y_(1)_y_(2))/(2))`
(1) `x^(2)+2y^(2)le1` represents interior region of ellipse `x^(2)+2y^(2)le1` as shown in the given figure.
Clearly, for any two points A and B in the shaded region, mid point of AB also lies in the shaded region.

(2) Max `{|x|,|y|}le1`
`rArr" "|x|le1 and |y|le1`
`rArr" "-1lexle1 and -1leyle1`
This represents the interior region of a square with its sides
`x=pm1 and y=pm1` as shown in the given figure.

(3) `x^(2)-y^(2)le1`
represents the exterior region of hyperbola as shown in the given figure.

As shown in the figure for points A and B selected in the region, the midpoint does not necessarily lie in the same region.
(4) `y^(2)lex` represents interior region of the parabola `y^(2)=x` as shown in the given figure.

Clearly, for any two points A and B in the shaded region, mid point of AB also lies in the shaded region.