Q206 / 270 · NDA 2019 (Phase 1) · Mathematics

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

  • A
    2
  • B
    -2
  • C
    4
  • Correct answer
    D
    -4

Solution

Given, opposite vertices of rectangle are $\mathrm{A}(1,3)$ and $\mathrm{C}(5,1)$ We know, diagonals of rectangle bisect each other. So, midpoint of AC lies on line $\mathrm{y}=2 \mathrm{x}+\mathrm{c}$. $\begin{aligned} & \text { Mid point of } \mathrm{AC}=\left(\frac{1+5}{2}, \frac{3+1}{2}\right)=\left(\frac{6}{2}, \frac{4}{2}\right)=(3,2) \\ & \mathrm{y}=2 \mathrm{x}+\mathrm{c} \Rightarrow 2=2(3)+\mathrm{c} \\ & \Rightarrow \mathrm{c}=2-6=-4\end{aligned}$

Hence, the correct answer is option: 'D'.

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