Question

The equation of the line segment AB is $y=x$. If A and B lie on the same side of the line mirror $2x-y=1$, the image of AB has the equation

• A. $x+y=2$
• B. $8x+y=9$
• C. $7x-y=6$
• D. none of these

Answer 1

Answer: (C)

$7x-y=6$

Let A(-1,-1) and B(0,0) be two points on the same side of the line $2x-y-1=0$
Let $(h,k)$ be the image of $A(-1,-1)$ w.r.t. line mirror $2x-y-1=0$
$\frac{h+1}{2}=\frac{k+1}{-1}=\frac{-2(-2)}{5}$
$\Rightarrow h=\frac{3}{5},k=-\frac{9}{5}$
Let $(h^{\prime },k^{\prime })$ be the image of $B(0,0)$ w.r.t. line mirror $2x-y-1=0$
$\frac{h^{\prime }}{2}=\frac{k^{\prime }}{-1}=\frac{-2(-1)}{5}$
$\Rightarrow h^{\prime }=\frac{4}{5},k^{\prime }=-\frac{2}{5}$
So, the image of $A(-1,-1)$ and $B(0,0)$ w.r.t the line $2x-y-1=0$ are $(\frac{3}{5},-\frac{9}{5})$ and $(\frac{4}{5},-\frac{2}{5})$
Hence the equation of image of AB is
$y+\frac{9}{5}=7(x-\frac{3}{5})$
$\Rightarrow 7x-y=6$