Question

# The points $$(1,3)$$ and$$(5,1)$$ are two opposite vertices of a rectangle. The other two vertices lie on the line $$y=2x+c$$. Find $$c$$ and the remaining vertices.

Solution

## $$m\left(\dfrac{5+1}{2}, \dfrac{3+1}{2}\right)$$$$\therefore m(3, 2)$$equation of $$'BD'$$ is $$y=2x+c$$$$\Rightarrow 2=6+c$$$$\therefore c=-4$$let $$(\alpha, \beta)$$$$\beta=2\alpha-4$$   ....(1)slope=$$2$$$$\Rightarrow \dfrac{\beta-3}{\alpha-1}\times \dfrac{\beta-1}{\alpha-5}=-1$$$$\Rightarrow (2\alpha-4-3)(2\alpha-4-1)=-(\alpha-1)(\alpha-5)$$$$\Rightarrow (2\alpha - 7) (2\alpha - 5) = -(\alpha - 1) (\alpha - 5)$$$$\Rightarrow 4\alpha^2-10\alpha-14\alpha+35=-\alpha^2+5\alpha+2=5$$$$\Rightarrow 5\alpha^2-30\alpha+40=0$$$$\Rightarrow \alpha^2-6\alpha+8=0$$$$\Rightarrow \alpha^2 - 4\alpha - 2\alpha + 8 = 0$$$$\Rightarrow \alpha(\alpha-4)-2(\alpha-4)=0$$$$\therefore \alpha=2, \alpha=4$$when $$\alpha=2$$, $$\beta=0$$when $$\alpha=4$$, $$\beta=4$$$$\therefore B(2, 0)$$ & $$D(4, 4)$$Mathematics

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