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  • The points (1, 3) and (5, 1) are two opposite vertices of a rectangle. The other two vertices lie on the line <img alt="\mathrm{y = 2x + c}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5C

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

Option: 1

(4,0) and (2,2)


Option: 2

(4,4) and (2,0)


Option: 3

(4,2) and (4,0)


Option: 4

None of the these 


Answers (1)

best_answer

Let \mathrm{A (1, 3) \; and \; C = (5, 1)}. The middle point P of AC is (3, 2). It lies on the other diagonal \mathrm{BD (y =2x + C)}

\mathrm{\therefore 2=2 \times 3+C \Rightarrow C=-4}

\therefore \mathrm{PB}=\mathrm{PD}=\frac{1}{2} \mathrm{AC}=\frac{1}{2} \sqrt{(5-1)^2+(1-3)^2}=\sqrt{5}

\mathrm{\tan \theta=2 \Rightarrow \sin \theta=\frac{2}{\sqrt{5}}\; \& \cos \theta=\frac{1}{\sqrt{5}}}

 coordinate of D, B are

\begin{aligned} & \equiv(3 \pm \sqrt{5} \cos \theta, 2 \pm \sqrt{5} \sin \theta) \\ \\& =(3 \pm 1,2 \pm 2)=(4,4) \text { and }(2,0) \end{aligned}

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Nehul

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