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Q.
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
Straight Lines
Solution:
$\because$ Diagonals of a rectangle bisect each other.
$\therefore$ Mid-point of $(1,3)$ and $(5,1)$ satisfies
$y-x+\lambda=0 .....$(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)
$\Rightarrow 2-3+\lambda=0 \Rightarrow \lambda=1$