For a point P in the plane, let d1( P ) and d2( 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 ≤ d1( P )+d2( P ) ≤ 4, is

Question

For a point P in the plane, let d1( P ) and d2( 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 ≤ d1( P )+d2( P ) ≤ 4, is  (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

2 ≤ d 1( p )+ d 2( p ) ≤ 4 For P (α, β), α>β ⇒ 2 √2 ≤ 2 α ≤ 4 √2 √2 ≤ α ≤ 2 √2 ⇒ Area of region =((2 √2)2-(√2)2) =8-2=6 sq. units <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/4db7630916611e622180411869be5e2a-.png" />

Q. For a point P in the plane, let d1​(P) and d2​(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≤d1​(P)+d2​(P)≤4, is ____

2≤d1​(p)+d2​(p)≤4 For P(α,β),α>β ⇒22​≤2α≤42​ 2​≤α≤22​ ⇒ Area of region =((22​)2−(2​)2) =8−2=6 sq. units

$2 \leq d_{1} (p) + d_{2} (p) \leq 4$
For $P (α, β), α > β$
$\Rightarrow 22 \leq 2 α \leq 42$
$2 \leq α \leq 22$
$\Rightarrow$ Area of region $= ((22)^{2} - (2)^{2})$
$= 8 - 2 = 6$ sq. units