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

If ∠BAO=θ  then  BM=2sinθ  and MO=BM=2sinθ,MA=2cosθ .                     Hence A=(2cosθ−2sinθ,0)  and B(−2sinθ,2sinθ)                                                              Since P ( x, y) is the mid point of AB ,                                                                                             2x=(2cosθ)+(−4sinθ)  or  cosθ−2sinθ=x                                                                          2y=(2sinθ)  or  sinθ=y                                                                                                                                                                                                  Eliminating θ , we have   (x+2y)2+y2=1  or ,x2+5y2+4xy−1=0