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Im trying to find radius of given circle below and its center coordinates. Base circle is unit circle with radius 1 as well as coordinates for p1 and p2 are given beforehand

enter image description here

Up to this point I know that $$ |p_1 - c| = r $$

$$ |p_2 - c| = r $$

$$ r^2 + 1 = c^2 $$

But somehow I got stuck to solve and figure out radius and center points of circle. Is there any thing am I missing?

hero_05
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    What exactly are you given? What you call the base circle seems to be the unit circle centered at the origin, but apart from that it's a bit unclear. Are you given the coordinates of $p_1$ and $c$, for instance? – Arthur Apr 06 '20 at 22:39
  • I edited question. Coordinates for p1 and p2 are given only, Need to find radius and coordinates for c – hero_05 Apr 06 '20 at 22:47

1 Answers1

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This is how I read the problem: there are 2 arbitrary points M and N with their Cartesian coordinates inside the circle $w$, find the circle $\alpha $ , including points M,N and radius $r$. There are 2 ways of problem's solution: geometrical and analytical.

  1. Geometrical method. enter image description here

The circles $w$ and $\alpha $ are called orthogonal. We need to inverse one of a given point relative to $w$, e.g. point$N$, and $N'$ is an inverse image of $N$. The construct of $N'$ is marked green. The circle $\alpha $ including $M$, $N$ and $N'$ is orthogonal to the base circle $w$ in conformance with inversion feature.

  1. Analytical method.

$\alpha $: $(x-xp)^2+(y-yp)^2=r^2$

Let us construct 3 equations in Cartesian coordinates having 3 variables $xp, yp, r$:

1) $(xm-xp)^2+(ym-yp)^2=r^2$

2) $(xn-xp)^2+(yn-yp)^2=r^2$

3) $ r^2+1=xp^2+yp^2$

Thus, there are 3 quadratic equations, which can be easily solved for 3 required variables $xp, yp, r$.

ole
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  • What's wrong? What do you mean? The equation seems to be correct. It specifies the orthogonality of circles. – ole Apr 07 '20 at 13:48