The points $(1,3),(5,1)$ are opposite vertices of a diagonal of a rectangle. If the other two vertices lie on the line $y=2 x+c$, then one of the vertex on the other diagonal is

• A

  (1,-2)
• B

  (0,-4)
• Correct answer
  C

  (2,0)
• D

  (3,2)

Solution

Diagonals of rectangle bisect each other. $\therefore \quad$ Midpoint of $(1,3)$ and $(5,1)$ is $(3,2)$. Also, $y=2 x+$ c passes through $(3,2)$. $\begin{array}{rlrl} \therefore & 2 & =2(3)+c \\ & \therefore & c & =-4 \end{array}$ $\therefore \quad$ Other two vertices lie on $y=2 x-4$. Let co-ordinates of $\mathrm{B}$ be $(x, y)$ i.e., $(x, 2 x-4)$ slope of $\mathrm{AB} \times$ slope of $\mathrm{BC}=-1$ $\begin{aligned} & \Rightarrow\left(\frac{2 x-4-3}{x-1}\right)\left(\frac{2 x-4-1}{x-5}\right)=-1 \\ & \Rightarrow\left(\frac{2 x-7}{x-1}\right)\left(\frac{2 x-5}{x-5}\right)=-1 \\ & \Rightarrow x^2-6 x+8=0 \\ & \Rightarrow x=4,2 \end{aligned}$ When $x=4, y=4$ When $x=2, y=0$ $\therefore \quad$ Vertex of the other diagonal is $(2,0)$.

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