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.
Taxicab Geometry: solution to $d(P, A) = 2 d(P, B)$ for two points, $A = (4,7)$ and $B = (5,4)$?
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geometry
metric-spaces
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4What have you tried? Do you know what the set of points at distance $2$ from $A$ looks like? – 2012-09-17
1 Answers
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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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0I already did that, I got 16 lineal equations, but i don´t know how to get the graph. Can you help me? – 2012-09-19