# Circle formula given two points and a manipulable radius

I need to find the formula for a specific circle. I know two points that is on the circle: (1,2) and (10,16)

I need to be able to manipulate the radius, so that I can find the formula that I am looking for. i.e: I will put this formula in excel, and then see what are the y values of x being 2 to 9 if I have a radius of 10 or 15 or 18 or 200.

Hope my question makes sense, I appreciate any help.

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What do you want to get from the formula? The center of the circle? –  bubba Aug 9 '13 at 10:26

The center of the circle can move along the line (perpendicular bisector) which is orthogonal to the connection between $(1;2)$ and $(10;16)$.

The general equation of a circle is:

$$(X - M_x)^2 + (Y - M_y)^2 = r^2$$

$$(1 - M_x)^2 + (2 - M_y)^2 = r^2$$ $$(10 - M_x)^2 + (16 - M_y)^2 = r^2$$

or

\begin{align}1 - 2M_x + M_x^2 + 4 - 4M_y + M_y^2 & = r^2\\ 100 - 20M_x + M_x^2 + 256 - 32M_y + M_y^2 & = r^2\end{align}

Subtracting the second equation from the first:

$$-99 + 18M_x - 252 +28M_y = 0$$

Hence, we get linear expression for $M_y$ you could use in Excel:

$$M_y = - \frac{18}{28}M_x + \frac{351}{28}$$

Squared radius $r^2$ can be determined from the circle equation: $$r^2 = (1 - M_x)^2 + (1 - M_y)^2$$

For $r = 10, 15, 18, 200$ the squared radius would be $100, 225, 324, 4000$.

\begin{aligned} r^2 & = (1 - M_x)^2 + (- \frac{323}{28} + \frac{18}{28}M_x)^2 \\ & = 1 - 2M_x + M_x^2 + \frac{104329}{784} - \frac{11808}{784}M_x + \frac{324}{784}M_x^2\\ & = \frac{324}{784}M_x^2 - \frac{11024}{784}M_x + \frac{105113}{784} \end{aligned}

This allows us to derive $M_x$ from $r^2$. Note that two solutions exist as the center of the circle can bei either on the right or on the left of $(9.5; 9)$.

Asking WolframAlpha reveals another formula suitable for Excel:

$$M_x = \frac{14\sqrt{1108r^2-93025} + 3299}{554}$$

To finally compute the $Y_i$ values corresponding to your $X_i$ values:

$$Y_i = M_y \pm \sqrt{r^2 - (X_i - M_x)^2}$$

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