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

Question

Infinity Learn · Accepted Answer

Suppose that the coordinatres of the point on the line $$x=2y$$ is $$2$$α,α and suppose that  the foot of the perpendciular from $$P2$$α,α on the line $$x=y$$ is  Qβ,β. Let Rh,k be the midpoint of PQ. Slope of PQ $$=k$$−αh−$$2$$α$$=$$−$$1$$, since PQ is perpendicualr to the line $$y=x$$⇒k−α$$=$$−$$h+2$$α⇒α$$=h+k3$$…………. (1)Also$$ $$2h=2$$α$$+$$β and $$2k=$$α$$+$$β⇒$$2h=$$α$$+2k$$⇒α$$=2h$$−$$2k$$…………. $$(2)From$$ $$(1) and $$(2)h+k3=2h$$−k∴ locus is $$6x$$−$$6y=x+y$$⇒$$5x=7y$$

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

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