Question

Two opposite vertices of a rectangle are $(1,3)$ and $(5,1)$. If the rest two vertices lie on the line $y-x+\lambda =0$, then $\lambda$ is equal to

• A. 1
• B. $-1$
• C. 2
• D. 3

Answer 1

Answer: (A)

$\because$ Diagonals of a rectangle bisect each other.
$\therefore$ Mid-point of $(1,3)$ and $(5,1)$ satisfies
$y-x+\lambda =\mathrm{0.....}$(i)
Mid-point of $(1,3)$ and $(5,1)$
$=\left(\frac{1+5}{2},\frac{3+1}{2}\right)=\left(\frac{6}{2},\frac{4}{2}\right)=(3,2)$
$\therefore$ Point $(3,2)$ satisfies Eq. (i)
$\Rightarrow 2-3+\lambda =0\Rightarrow \lambda =1$