Question
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 $$\left( {{x_1},\,{y_1}} \right)$$ and $$\left( {{x_2},\,{y_2}} \right)$$ in the
the region, the point $$\left( {\frac{{{x_1} + {x_2}}}{2},\,\frac{{{y_1} + {y_2}}}{2}} \right)$$ is also in the region. The inequality defining this region is :
A.
$${x^2} + 2{y^2} \leqslant 1$$
B.
$${\text{max }}\left\{ {\left| x \right|,\left| y \right|} \right\} \leqslant 1$$
C.
$${x^2} - {y^2} \leqslant 1$$
D.
$${y^2} - {x^2} \leqslant 0$$
Answer :
$${x^2} - {y^2} \leqslant 1$$
Solution :
Option A, $${x^2} + 2{y^2} \leqslant 1$$
It can be written as $$\frac{{{x^2}}}{1} + \frac{{{y^2}}}{{\frac{1}{2}}} \leqslant 1$$
which represents inside region of an ellipse
We know for an ellipse, the mid-point of any two points in the region, is also in the region.
Let's take two points $$\left( {0,\,0} \right)$$ and $$\left( {\frac{1}{2},\,0} \right)$$ which lies in the region.
Now, mid-point is $$\left( {\frac{1}{4},\,0} \right)$$ which also lies in the region.
Now, option B, $${\text{max }}\left\{ {\left| x \right|,\left| y \right|} \right\} \leqslant 1$$
Now, option C, $${x^2} - {y^2} \leqslant 1$$
which represents inside region of hyperbola.
A hyperbola has two parts . If we take 2 points , one in one part and other in other part, the mid-point need not to be inside the hyperbola.
Let's take $$\left( { - \frac{1}{2},\,0} \right)$$ and $$\left( {\frac{1}{4},\,0} \right)$$ as two points.
Their mid-point is $$\left( { - \frac{1}{8},\,0} \right)$$ is not in the region of hyperbola.
Hence, option C does not satisfy property P.
Option D, $${y^2} - {x^2} \leqslant 0$$
which represents inside region of pair of straight lines.
Mid-point of any two point in the region will lie in the region only.
Hence, satisfies property P.