Circle O with center (x,y), passes through the points A(0,0), B(-3,0), and C(1, 2). Find the coordinates of the center of the circle I

Question

Circle O with center (x,y), passes through the points A(0,0), B(-3,0), and C(1, 2). Find the coordinates of the center of the circle
In your final answer, include all formulas and calculations used to find Point O, x,y). the center of circle O.

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4 months 2022-01-07T02:37:23+00:00 1 Answer 0

The center has the coordinates $$(-1.5, 2)$$ and the radius 2.5

Step-by-step explanation:

Let $$(x_0,y_0)$$ be the center of the circel and r be the radius, then the equation of the circle is

$$(x-x_0)^2+(y-y_0)^2=r^2$$

Circle O with center $$(x_0,y_0)$$ passes through the points A(0,0), B(-3,0), and C(1, 2), so

$$\begin{array}{l}(0-x_0)^2+(0-y_0)^2=r^2\\ \\(-3-x_0)^2+(0-y_0)^2=r^2\\ \\(1-x_0)^2+(2-y_0)^2=r^2\end{array}\Rightarrow \begin{array}{l}x_0^2+y_0^2=r^2\\ \\(3+x_0)^2+y_0^2=r^2\\ \\(1-x_0)^2+(2-y_0)^2=r^2\end{array}$$

Subtract from the second equation the first one:

$$(3+x_0)^2+y_0^2-x_0^2-y_0^2=r^2-r^2\\ \\(3+x_0)^2-x_0^2=0\\ \\(3+x_0)^2=x_0^2\\ \\3+x_0=x_0\ \text{or}\ 3+x_0=-x_0\\ \\3=0\ \text{or}\ 2x_0=-3,\ x_0=-1.5$$

Substitute it into the last two equations:

$$\begin{array}{l}(3-1.5)^2+y_0^2=r^2\\ \\(1+1.5)^2+(2-y_0)^2=r^2\end{array}\Rightarrow \begin{array}{l}2.25+y_0^2=r^2\\ \\6.25+(2-y_0)^2=r^2\end{array}$$

Subtract them:

$$6.25+(2-y_0)^2-2.25-y_0^2=r^2-r^2\\ \\4+4-4y_0+y_0^2-y_0^2=0\\ \\4y_0=8\\ \\y_0=2$$

Substitute into the first equation:

$$(-1.5)^2+2^2=r^2\\ \\r^2=2.25+4\\ \\r^2=6.25\\ \\r=2.5$$

So, the center has the coordinates $$(-1.5, 2)$$ and the radius 2.5