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For a point P in the plane, let d(1)(P) ...

For a point P in the plane, let `d_(1)(P) and d_(2)(P)` be the distances of 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 le d_(1)(P) + d_(2)(P) le 4`, is ________.

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The correct Answer is:
6
For P(x,y), we have
`2led_(1)(P)+d_(2)(P)le4`
`rArr" "2le(|x-y|)/(sqrt(2))+(|x+y|)/(sqrt(2))le4`
`rArr" "2sqrt(2)le |x-y|+|x+y|le4sqrt(2)`
IN first quadrant if `xgty,` we have
`2sqrt(2)lex-y+x+yle4sqrt(2)`
`"or "sqrt(2)lexle2sqrt(2)`
The region of points satisfying these inequalities is

In first quadrant if `xlty`, we have
`2sqrt(2)ley-x+x+yle4sqrt(2)`
`"or "sqrt(2)leyle2sqrt(2)`
The region of points satisfying these inequalities is

Combining above two regions we have

Area of the shaded region `=((2sqrt(2))^(2)-(sqrt(2))^(2))`
`=8-2=6` sq. units
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