# Taxicab Geometry: solution to $d(P, A) = 2 d(P, B)$ for two points, $A = (4,7)$ and $B = (5,4)$?

I need the solution for $A= (4,7)$, $B= (5,4)$, please also the graph and an explanation about the procedure for getting the graph.

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What have you tried? Do you know what the set of points at distance $2$ from $A$ looks like? –  Ross Millikan Sep 17 '12 at 2:07

Let $P(x,y)$ be the point satisfying $d(P,A)=2d(P,B)$

$d(P,A)=|x-4|+|7-y|$

$d(P,B)=|x-5|+|4-y|$

$d(P,A)=2d(P,B)\implies |x-4|+|7-y|=2|x-5|+2|4-y|$

Now, check solutions on $X\times Y$ where

$X=\{(-\infty,4),[4,5],(5,\infty)\}$

$Y=\{(-\infty,4),[4,7],(7,\infty)\}$

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I already did that, I got 16 lineal equations, but i don´t know how to get the graph. Can you help me? –  Pedro Vergara Sep 19 '12 at 2:10