A variable line ' L ' is drawn through O(0,0) to meet the  lines L1:y−x−10=0 and L2:y−x−20=0 at the  points A and B respectively. A point P is taken on ' L′ such that 2OP=1OA+1OB . Locus of ' P′ is

Let the parametric equation of drawn line isxcos⁡θ=ysin⁡θ=r⇒ x=rcos⁡θ,y=rsin⁡θ Putting it in ' L1 ', we get rsin⁡θ=rcos⁡θ+10⇒ 1OA=sin⁡θ−cos⁡θ10Similarly, putting the general point of drawn line is the equation of L2, we get 1OB=sin⁡θ−cos⁡θ20 Let P=(h,k) and OP=r⇒rcos⁡θ=h,rsin⁡θ=k , we have 2r=sin⁡θ−cos⁡θ10+sin⁡θ−cos⁡θ20⇒ 40=3rsin⁡θ−3rcos⁡θ⇒ 3y−3x=40