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For a point P in the plane, let d1(P)a n...

For a point `P` in the plane, let `d_1(P)a n dd_2(P)` be the distances of the point `P` from the lines `x-y=0a n dx+y=0` respectively. The area of the region `R` consisting of all points `P` lying in the first quadrant of the plane and satisfying `2lt=d_1(P)+d_2(P)lt=4,` is

Text Solution

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PLAN Distance of a point `(x_(1),y_(1))` from `ax+by+c=0` is given by
`|(ax_(1)+by_(1)+c)/(sqrt(a^(2)+b^(2)))|`

Let `P(x,y)` is the point in first quadrant.
Now, `2 le |(x-y)/(sqrt(2))|+|(x+y)/(sqrt(2))| le 4`
`2sqrt(2) le |x-y|+|x+y| le 4sqrt(2)`
Case I `x ge y`
`2sqrt(2) le (x-y)+(x+y) le 4sqrt(2)implies x in [sqrt(2),2sqrt(2)]`
Case II `x lt y`
`2sqrt(2) le y-x+(x+y) le 4sqrt(2)`
`y in [sqrt(2),2sqrt(2)]`
`impliesA=(2sqrt(2))^(2)-(sqrt(2))^(2)=6` sq units
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