Question

The points $(1,3)$ and $(5,1)$ are the opposite vertices of a rectangle. The other two vertices lie on the line $y=2x+c$, then the value of $c$ will be
A. $4$
B. $-4$
C. $2$
D. $-2$

Answer 1

Hint: In this question, we are to find the value of $c$ i.e., the y-intercept of the given line. To find this, the property of a rectangle that the diagonals of a rectangle bisect each other at the mid-point is used. The midpoint of the diagonal is to be calculated and then substituted in the given line to find the $c$ value.

Formula used: The midpoint of a line joining of $({{x}_{1}},{{y}_{1}})$ and $({{x}_{2}},{{y}_{2}})$ is
$=\left( \dfrac{{{x}_{1}}+{{x}_{2}}}{2},\dfrac{{{y}_{1}}+{{y}_{2}}}{2} \right)$
 The slope-intercept form of a line is
$y=mx+c$

Complete step by step solution: Given that,
The two opposite vertices of a rectangle are $A(1,3)$ and $B(5,1)$
The equation of the other diagonal $\overleftrightarrow{BD}$ is $y=2x+c$
In a rectangle, the diagonals are bisecting each other. So, their point of intersection is their mid-point.
So, midpoint of the diagonal $\overleftrightarrow{AC}$ is
$\begin{align}
  & =\left( \frac{1+5}{2},\frac{3+1}{2} \right) \\
 & =\left( \frac{6}{2},\frac{4}{2} \right) \\
 & =(3,2) \\
\end{align}$
Thus, the other diagonal $\overleftrightarrow{BD}$ also passes through this point.
So, substituting this point in the equation of the diagonal $\overleftrightarrow{BD}$, we get
$\begin{align}
   y=2x+c \\
  \Rightarrow 2=2(3)+c \\
  \Rightarrow c=2-6 \\
  \therefore c=-4 \\
\end{align}$

Thus, Option (B) is correct.

Note: Here we need to remember that the diagonals of a rectangle bisect each other. By using this property, the value is obtained. We can also use another method in which, using the given points of the diagonal its equation is framed. Since the two diagonals are perpendicular to each other, the product of the slopes of the two diagonals is used for finding the value of $c$.