Question

Two points (a, 3) and (5, b) are the opposite vertices of a rectangle. If the other two vertices lie on the line $y=2x+c$ which also passes through the point $(a/c,b/c)$ then what is the value of c?

• 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)

The correct options are
A $2\sqrt{2}+1$
C $1-2\sqrt{2}$

As the midpoint of the line joinning the given two opposite coordinates of the rectangle is on the the line $y=2x+c$

$\frac{b+3}{2}=2\left(\frac{a+5}{2}\right)+c\Rightarrow 2a+2c-b+7=0$
$\frac{b}{c}=2\frac{a}{c}+c\Rightarrow 2a-b=-c^2$
So, $c^2-2c-7=0$
$\Rightarrow c=1\pm 2\sqrt{2}$