Question

Two points $(a,3)$ and $(5,b)$ are the opposite vertices of a rectangle. If the coordinates $(x,y)$ of the other two vertices satisfy the relation $y=2x+c$ where $c^2+2a-b=0$ then the value $c$ can be

• A. $2\sqrt{2}+1$
• B. $2\sqrt{2}-1$
• C. $1-2\sqrt{2}$
• D. $-1-2\sqrt{2}$

Answer 1

Answer: (A), (C)

$2\sqrt{2}+1$
$1-2\sqrt{2}$

Mid-point of the other vertices is also the mid-point of the given vertices and hence satisfies the given relation
So, $\frac{b+3}{2}=2\left(\frac{a+5}{2}\right)+c$
$\Rightarrow 2a+2c-b+7=0$
Also, $c^2+2a-b=0$
$\Rightarrow c^2-2c-7=0\Rightarrow c=1\pm 2\sqrt{2}$