Question

For a point $P$ in the plane, let $d_{1}( P )$ and $d_{2}( P )$ be the distances of the point $P$ from the lines $x-y=0$ and $x+$ $y=0$ respectively. The area of the region $R$ consisting of all points $P$ lying in the first quadrant of the plane and satisfying $2 \leq d_{1}( P )+d_{2}( P ) \leq 4$, is ____

Answer 1

$2 \leq d _{1}( p )+ d _{2}( p ) \leq 4$
For $P (\alpha, \beta), \alpha>\beta$
$\Rightarrow 2 \sqrt{2} \leq 2 \alpha \leq 4 \sqrt{2}$
$\sqrt{2} \leq \alpha \leq 2 \sqrt{2}$
$\Rightarrow $ Area of region $=\left((2 \sqrt{2})^{2}-(\sqrt{2})^{2}\right) $
$=8-2=6$ sq. units