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# Help with coordinates

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Three Points on a circle: (-17,-2), (0,15), (9,-2). What is the equation of the circle?

Jun 5, 2021

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The equation for a circle is $$(x-a)^2+(y-b)^2=r^2$$, where (a, b) is the center, and r is the radius.

Plugging in (x, y) = (-17, -2) into the above equation, we have $$(-17-a)^2+(-2-b)^2=r^2$$

Plugging in (x, y) = (0, 15) into the above equation, we have $$a^2+(15-b)^2=r^2$$

Plugging in (x, y) = (9, -2) into the above equation, we have$$(9-a)^2+(-2-b)^2=r^2$$.

In this, we have 3 variables (a, b, and r), and 3 equations, and hence can solve for each of them.  A bit of algebra later,

a = -4, b = 2, and r = $$\sqrt{185}$$.

Substituting these values into the equation for the circle,

The equation becomes $$\left(x+4\right)^{2}+\left(y-2\right)^{2}=185$$

Jun 5, 2021