- Home
- Standard 11
- Mathematics
9.Straight Line
normal
For a point $P$ in the plane, let $d_1(P)$ and $d_2(P)$ be the distance 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
A
$4$
B
$5$
C
$6$
D
$7$
(IIT-2014)
Solution
let $ p(h, k) $
$2 \leq\left|\frac{h-k}{\sqrt{2}}\right|+\left|\frac{h+k}{\sqrt{2}}\right| \leq 4 $
$\Rightarrow \quad 2 \sqrt{2} \leq|h-k|+|h+k| \leq 4 \sqrt{2} $
$\text { if } \quad h \geq k$
$\Rightarrow \quad 2 \sqrt{2} \leq x-y+x+y \leq 4 \sqrt{2} \quad$ or $\quad \sqrt{2} \leq x \leq 2 \sqrt{2}$
similarly when $k > h$
we have $\sqrt{2} \leq y \leq 2 \sqrt{2}$
The required area $=(2 \sqrt{2})^2-(\sqrt{2})^2=6$.
Standard 11
Mathematics
