watch5826c

2022-06-24

Given a point $({x}_{0},{y}_{0})$ and a radius r, how do you find the set of all circles that have that radius that pass through the point?

lorienoldf7

Beginner2022-06-25Added 19 answers

Step 1

We are interested in circles in the $\u27e8x,y\u27e9$ plane, therefore they will have Cartesian equation:

$(x-{x}_{C}{)}^{2}+(y-{y}_{C}{)}^{2}={r}^{2}$

where their radius $r>0$ is fixed, so it remains to understand how to determine the centers.

In particular, given that we want the circles in question all to pass through $({x}_{P},\phantom{\rule{thinmathspace}{0ex}}{y}_{P})$ and at the same time we have radius $r>0$ , it's evident that $({x}_{C},\phantom{\rule{thinmathspace}{0ex}}{y}_{C})$ must belong to the circle with center $({x}_{P},\phantom{\rule{thinmathspace}{0ex}}{y}_{P})$ and radius $r>0$ , that is:

$\{\begin{array}{l}{x}_{C}={x}_{P}+r\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}(u)\\ {y}_{C}={y}_{P}+r\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}(u)\end{array}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{thickmathspace}{0ex}}u\in [0,\phantom{\rule{thinmathspace}{0ex}}2\pi )\phantom{\rule{thinmathspace}{0ex}}.$

In this way, we have determined the Cartesian equation of the sheaf of circles obtainable according to the chosen value of $u\in [0,\phantom{\rule{thinmathspace}{0ex}}2\pi )$

We are interested in circles in the $\u27e8x,y\u27e9$ plane, therefore they will have Cartesian equation:

$(x-{x}_{C}{)}^{2}+(y-{y}_{C}{)}^{2}={r}^{2}$

where their radius $r>0$ is fixed, so it remains to understand how to determine the centers.

In particular, given that we want the circles in question all to pass through $({x}_{P},\phantom{\rule{thinmathspace}{0ex}}{y}_{P})$ and at the same time we have radius $r>0$ , it's evident that $({x}_{C},\phantom{\rule{thinmathspace}{0ex}}{y}_{C})$ must belong to the circle with center $({x}_{P},\phantom{\rule{thinmathspace}{0ex}}{y}_{P})$ and radius $r>0$ , that is:

$\{\begin{array}{l}{x}_{C}={x}_{P}+r\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}(u)\\ {y}_{C}={y}_{P}+r\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}(u)\end{array}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{thickmathspace}{0ex}}u\in [0,\phantom{\rule{thinmathspace}{0ex}}2\pi )\phantom{\rule{thinmathspace}{0ex}}.$

In this way, we have determined the Cartesian equation of the sheaf of circles obtainable according to the chosen value of $u\in [0,\phantom{\rule{thinmathspace}{0ex}}2\pi )$

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