Question

$P$ is a point which moves in the $X Y$ plane such that the point $P$ is nearer to the centre of the square than any of the sides. The four vertices of the square are $( \pm a , \pm a )$. The region in which $P$ will move is bounded by parts of parabola of which the equation is

• A. $y^2=a^2-2 a x$
• B. $y=a^2+2 ax ^2$
• C. $x=a^2-2 a y^2$
• D. $x^2=a^2+2 a y$

Answer 1

Answer: (A), (D)

If $P ( x , y )$, then according to the question,
$\sqrt{ x ^2+y^2}<|x-a|, \sqrt{x^2+y^2}< |x+a|$
$\sqrt{x^2+y^2}<|a-y|, \sqrt{x^2+y^2}< |a+y|$
On simplification, the region bounded by these curves becomes
$y^2= a ^2-2 ax , y ^2= a ^2+2 a x, x ^2= a ^2-2 a y \text { and } x ^2= a ^2+2 ay$