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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 $$\alpha=4$$, $$\beta=4$$
$$\therefore B(2, 0)$$ & $$D(4, 4)$$

1232962_1189231_ans_4fe43d093aed481db5fe983570095285.png

Mathematics

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