A variable line whose slope is -2 cuts x and y-axes respectively at points A and C. If Rhombus ABCD is completed such that the vertex B lies on the line y = x , then the locus of the vertex D is

Let the line be y=−2x+c Let B be (α,α) and D be (h,k)A≡(c2,0) and C≡(0,c)Comparing midpoints of AC and BD ∴ h+α=c4 and k+α=c2 Eliminating c,  2h+2α=k+α or  k−2h=α-----(1) Also AC⊥BD α−kα−h×(−2)=−1∴ 2α−2k=α−h∴ 2k−h=α----(2) From (1) and (2), k−2h=2k−h or x+y=0, which is required locus