Question

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

Answer 1

Since, diagonals of rectangle bisect each other, so mid point of $(1,3)$ and $(5,1)$ must satisfy $y=2x+c$, i.e. $(3,2)$ lies on it.

$\Rightarrow 2=6+c\Rightarrow c=-4$
$\therefore$ Other two vertices lies on $y=2x-4$
Let the coordinate of $B$ be $(x,2x-4)$.
$\therefore$ Slope of $AB$ . Slope of $BC=-1$
$\Rightarrow \left(\frac{2x-4-3}{x-1}\right).\left(\frac{2x-4-1}{x-5}\right)=-1$
$\Rightarrow (x^2-6x+8)=0$
$\Rightarrow x=4,2$
$\Rightarrow y=4,0$
Hence, required points are $(4,4),(2,0)$.