The locus of the mid-points of the perpendiculars drawn from points on the line, x=2y to the line x = y is:

Suppose that the coordinatres of the point on the line x=2y is 2α,α and suppose that  the foot of the perpendciular from P2α,α on the line x=y is  Qβ,β. Let Rh,k be the midpoint of PQ. Slope of PQ =k−αh−2α=−1, since PQ is perpendicualr to the line y=x⇒k−α=−h+2α⇒α=h+k3…………. (1)Also 2h=2α+β and 2k=α+β⇒2h=α+2k⇒α=2h−2k…………. (2)From (1) and (2)h+k3=2h−k∴ locus is 6x−6y=x+y⇒5x=7y