A line segment of fixed length 2 units moves so that its ends are on the positive x-axis and on the part of the linex+y=0 which lies in the second quadrant. Then the locus of the midpoint of the line has the equations

As shown in the figure, line segment AB, of fixed length 2 units, slides such that one of its ends A is on x-axis and other ends A is x-axis and other end B is on line x+y=0Let Ph,k be midpoint of AB.Also, Let ∠BAO=θ.So, in triangle AMB,​ BM=2sinθ and AM=2cosθ.In triangle BMO, MO=MB=2sinθ∴   A≡2cosθ−2sinθ,0 and B≡−2sinθ,2sinθNow, Ph,k is the midpoint of AB.∴  2h=2cosθ−2sinθ+−2sinθ and 2k=2sinθ⇒   h=cosθ−2sinθ and k=sinθSquaring and adding, we get h+2k2+k2=1 or x2+5y2+4xy−1=0This is the required locus.