First slide

Hyperbola in conic sections

Question

$Each of the four inequalities given below defines a region in the xy plane. One of these four regions does not have$
$the following property . For any two points (x_{1}, y_{1}) and$ $(x_{2}, y_{2}) in the region, the point ((x_{1} + x_{2}) / 2, (y_{1} + y_{2}) / 2) is$
$also in the region. The inequality defining this region is$

Moderate

A

$x^{2} + 2 y^{2} \leq 1$
B

$max {| x |, | y |} \leq 1$
C

$x^{2} - y^{2} \leq 1$
D

$y^{2} - x \leq 0$

Solution

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

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