Question

In the upper-plane plane model for hyperbolic geometry, calculate the distance between the points A(0, 4) text{and} B(3, 5). Give your answer accurate to three decimals. Hint: Recall the definition of distance in the upper-half plane model.

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ANSWERED
asked 2021-02-10
In the upper-plane plane model for hyperbolic geometry, calculate the distance between the points \(A(0,\ 4)\ \text{and}\ B(3,\ 5).\) Give your answer accurate to three decimals. Hint: Recall the definition of distance in the upper-half plane model.

Answers (1)

2021-02-11
Step 1
We have the two points \(A (0,\ 4)\ \text{and}\ B (3,\ 5)\) let
\(\displaystyle{A}={\left({0},{4}\right)}\to{\left({x}_{{1}},{x}_{{2}}\right)}\)
\(\displaystyle\therefore{x}_{{1}}={0},{x}_{{2}}={4}\)
\(\displaystyle{B}={\left({3},{5}\right)}\to{\left({y}_{{1}},{y}_{{2}}\right)}\)
\(\displaystyle\therefore{y}_{{1}}={3},{y}_{{2}}={5}\)
We have distance,
\(\displaystyle{d}{i}{s}{\left(\begin{matrix}{x}_{{1}}&{y}_{{1}}\\{x}_{{2}}&{y}_{{2}}\end{matrix}\right)}\)
\(\displaystyle={2} \ln{{\left(\frac{{\sqrt{{{\left({x}_{{2}}-{x}_{{1}}\right)}^{2}+{\left({y}_{{2}}-{y}_{{1}}\right)}^{2}}}+\sqrt{{{\left({x}_{{2}}-{x}_{{1}}\right)}^{2}+{\left({y}_{{2}}-{y}_{{1}}\right)}^{2}}}}}{{{2}\sqrt{{{y}_{{1}}{y}_{{2}}}}}}\right)}}\)
Substitute the values
\(\displaystyle={2} \ln{{\left(\frac{{\sqrt{{{\left({4}-{0}\right)}^{2}+{\left({5}-{3}\right)}^{2}}}+\sqrt{{{\left({4}-{0}\right)}^{2}+{\left({5}-{3}\right)}^{2}}}}}{{{2}\sqrt{{{3}\times{5}}}}}\right.}}\)
Step 2
\(\displaystyle={2} \ln{{\left(\frac{{\sqrt{{{4}^{2}+{3}^{2}}}+\sqrt{{{4}^{2}+{8}^{2}}}}}{{{2}\sqrt{{15}}}}\right)}}\)
\(\displaystyle={2} \ln{{\left(\frac{{\sqrt{{{16}+{9}}}+\sqrt{{{16}+{64}}}}}{{{2}\sqrt{{15}}}}\right)}}\)
\(\displaystyle={2} \ln{{\left(\frac{{\sqrt{{{25}}}+\sqrt{{{80}}}}}{{{2}\sqrt{{15}}}}\right)}}\)
\(\displaystyle={2} \ln{{\left(\frac{{{5}+\sqrt{{{80}}}}}{{{2}\sqrt{{15}}}}\right)}}\)
\(\displaystyle\sqrt{{80}}=\sqrt{{{16}\times{5}}}\)
\(\displaystyle=\sqrt{{16}}\times\sqrt{{5}}\)
\(\displaystyle={4}\sqrt{{5}}\)
\(\displaystyle={2} \ln{{\left(\frac{{{5}+{4}\sqrt{{5}}}}{{{2}\sqrt{{15}}}}\right)}}\)
\(\displaystyle={2} \ln{{\left(\frac{13.94407191}{{7.7459666924}}\right)}}\)
\(= 1.18355406\)
\(= 1.184\)
So the answer is \(1.184\)
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