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

Question

Infinity Learn · Accepted Answer

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+2$$α$$=k+$$α or  k−$$2h=$$α$$-----(1) Also AC⊥BD α−kα−h×$$($$−$$2)=$$−$$1$$∴ $$2$$α−$$2k=$$α−h∴ $$2k$$−$$h=$$α$$----(2)$$ From (1) and (2), k−$$2h=2k$$−h or $$x+y=0$$, which is required locus

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

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material