Let P( , ) and Q( , ) be two points that lie on the curve ^2(x+y)+ ^2(x+y)+y^2+2 y=0 in the X Y -plane. If the distance between P and Q is d , then d=

Step-by-step solution

We have, ^2(x+y)+ ^2(x+y)+y^2+2 y=0 aligned & ^2(x+y)-1+ ^2(x+y)+y^2+2 y=0 & ^2(x+y)+ ^2(x+y)+y^2+2 y+1=2 aligned Now, as minimum value of ^2(x+y)+ ^2(x+y) is 2. aligned & x+y=0 and y^2+2 y+1=0 & x=-y and (y+1)^2=0 & x=-y and y=-1 & aligned Thus, x=1 and y=-1 Hence, the points P and Q coincides, and so d=0 Now, again if we take x+Y= and y^2+2 y+1=0 , then x= +1 and y=-1 Similarly, if x+y=2 and y^2+2 y+1=0 Then, x=2 +1, y=-1 d=