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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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Answer:

The center of the circle is

[tex](-\frac{3}{2},2)[/tex]

Step-by-step explanation:

Let the equation of the circle be

[tex]x^2+y^2+2ax+2by+c=0[/tex], where (-a,-b) is the center of this circle.

The points lying the circle must satisfy the equation of this circle.

A(0,0)

We substitute this point to get;

[tex]0^2+0^2+2a(0)+2b(0)+c=0[/tex]

[tex]\implies c=0[/tex]

B(-3,0)

[tex](-3)^2+0^2+2a(-3)+2b(0)+c=0[/tex]

[tex]\implies 9+0-6a+0+c=0[/tex]

[tex]\implies -6a+c=-9[/tex]

But c=0

[tex]\implies -6a=-9[/tex]

[tex]\implies a=\frac{3}{2}[/tex]

C(1,2)

[tex]1^2+2^2+2a(1)+2b(2)+c=0[/tex]

[tex]1+4+2a+4b+c=0[/tex]

[tex]2a+4b+c=-5[/tex]

Put the value of 'a' and 'c' to find 'b'

[tex]2(\frac{3}{2})+4b+0=-5[/tex]

[tex]3+4b+0=-5[/tex]

[tex]4b=-5-3[/tex]

[tex]4b=-8[/tex]

b=-2

Hence the center of the circle is

[tex](-\frac{3}{2},2)[/tex]

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