1)

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   2d1(P)+d2(P)4.   is   


A) 4

B) 5

C) 6

D) 2

Answer:

Option C

Explanation:

Plan Distance of a point (x1,y1) from  ax+bx+c=0 is given by

 |ax1+by1+ca2+b2|

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Let P(x,y)  is the point in first quadrant.

 Now,   2|xy2|+|x+y2|4

 22|xy|+|x+y|42

 

 Case I    xy

    22(xy)+(x+y)42

         x(2,22)

Case II     x<y

22yx+(x+y)42

  y(2,22)

        A=(22)2(2)2=6