1just an observation s is symmetrical in the axes.dunno what to do frm here.
Let S = \left\{(x,y): |||x|-2|-1|+|||y|-2|-1|=1 \right\}. If a wire is made out of S, then the length of such wire is?
Source : FIITJEE PT II (P1) Q.1 (Maths)
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1708\hspace{-16}$Here $\mid \mid \mid x\mid -2\mid -1\mid+\mid \mid \mid y\mid -2\mid -1\mid =1$\\\\\ Now First we will find The bound for $x$ and $y$\\\\ So $0 \leq \mid \mid \mid x\mid -2\mid -1\mid\leq 1$ and $0 \leq \mid \mid \mid x\mid -2\mid -1\mid\leq 1$\\\\ So $-1 \leq \mid \mid x\mid -2\mid -1 \leq 1$\\\\ $0 \leq \mid \mid x\mid -2\mid \leq 2$\\\\ $ -2\leq \mid x\mid -2\leq 2$\\\\ $0\leq \mid x \mid\leq 4$\\\\ So $-4\leq x\leq 4$\\\\ Similarly $-4\leq y\leq 4$\\\\ So $x\;,y \in\left[-4\;,4\right]$\\\\ Now Given expression is Symmetrical about both axis.\\\\ *(Which is Written by Rashab)\\\\ So We Will calculate in First - Quadrant and Symmetrical about Both Axis.\\\\ So First Quadrant::$x>0\;,y>0$\\\\ $\mid \mid x-2 \mid -1\mid +\mid \mid x-2 \mid -1\mid=1$\\\\ Now We have take Four Different cases:\\\\
\hspace{-16}(A):: $\;If $x>2$ and $y>2$\\\\ $\mid x-3 \mid +\mid y-3 \mid =1$\\\\ (B):: If $x<2$ and $y>2$\\\\ $\mid x-1 \mid +\mid y-3 \mid =1$\\\\ (C):: If $x<2$ and $y<2$\\\\ $\mid x-1 \mid +\mid y-1 \mid =1$\\\\ (D):: If $x>2$ and $y<2$\\\\ $\mid x-3 \mid +\mid y-1 \mid =1$\\\\ So We Draw Graph on First Quadrant and Then take Symmetry\\\\ on all $4$ Quadrant
\hspace{-16} $So Here are Total $4$ square and each side Length is =$\sqrt{2}$\\\\ So For all $4$ square side Length is $=16\sqrt{2}$\\\\ Now Same as $2^{nd}\;,3^{rd}\;,4^{th}$ Quadrant\\\\ So Total Length is = $4\times 16\sqrt{2}=64\sqrt{2}$
