Question

For points $P\equiv (x_1,y_1)$ and $Q\equiv (x_2,y_2)$ of the coordinate plane, a new distance $d(P,Q)=|x_1-x_2|+|y_1-y_2|$
Let $O\equiv (0,0),A\equiv (2,3)$ and $C\equiv (4,3)$ are four fixed points on the $x-y$ plane.
On the basis of the above information, answer the following questions:

Let $R(x,y)$ such that $R$ is equidistant from the points $O$ and $A$ with respect to new distance and if $0\le x<1$ and $0\le y<2$ then $R$ lie on a line segment whose equation is

• A. $x+y=3$
• B. $x+2y=3$
• C. $2x+y=3$
• D. $2x+2y=3$

Answer 1

Answer: (D)

$2x+2y=3$

$\because OR=AR$
$\Rightarrow |x-0|+|y-0|=|x-1|+|y-2|$
$\Rightarrow \left|x\right|+\left|y\right|=|x-1|+|y-2|$
$\because 0\le x<1$ and $0\le y<2$
$\therefore x+y=-(x-1)-(y-2)$
$\Rightarrow 2x+2y=3$Hence R lie on the line segment $2x+2y=3$