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.

Answer 1

Now, there are several ways to solve this type of circumference passing through three points exercises, let's consider the generic equation:
\[x^2+y^2+Dx+Ey+F=0 \]
Corresponds to the equation of the circle with the given center, since points A, B and C will be points belonging to this equation since we are given that C intersects these points it's just a matter of plotting their coordinates:
\[(0)^2+(y^2)+D(0)+E(0)+F=0\]
\[(-3)^2+(0)^2+D(-3)+E(0)+F=0 \]
\[(1)^2+(2)^2+D(1)+E(2)+F=0 \]
And now you have a system of equations composed by three variables and three equations, once you solve for the variables you'll obtain the generic form of the equation of the circle.
\[F=0\]
\[9-3D+F=0\]
\[1+4+D+2E+F=0 \]

Answer 2

that's certainly a valid approach.

Starting with the more familiar equation of a circle of radius r centered at (a,b) might be less intimidating.

This equation of a circle is \(x-h)^2 + (y-k)^2 =1

Answer 3

@mathmale what do k and h stand for in the equation?

Answer 4

Or, you can take a pair of points - any pair...

1) Find the midpoint of the line segment between them.
2) Find the equation of the line perpendicular to the segment at the midpoint.

Do that again with a different pair of points.

Find the intersection of the two lines so constructed. You have your center, (h,k).

Unique Results don't care how you find them.

Answer 5

Substitute the coordinates of each of the given points into that equation. This will give you 3 relationships, sufficient info with which to determine the center of the circle.

Answer 6

(h,k) would be the unknown center of the circle.

Answer 7

tkhunny's approach is valid and very interesting.

Answer 8

@tkhunny ok, so do I just find the midpoints ok the points?

Answer 9

No, follow the construction carefully. So far, we have three methods that need a lot of algebra. Be very careful.

Answer 10

Well, now that we are speaking of methods, it is also valid to construct the triangle \(\triangle ABC\) which are the given belonging to the circumference.
Finding the circumcenter of triangle \(\triangle ABC\) should suffice to find the center of the circumference.

Answer 11

@owlcoffee: another interesting alternative approach! Be aware that not many OpenStudy users would recognize the word "circumcenter," let alone know its meaning.

Answer 12

@owlcoffee Well, that's really the same as mine. You were just more honest about it.

@Elizbeth0197 If anyone EVER tries to tell you there is only one way to proceed, feel free to chuckle at them. Be polite. :-)

Answer 13

I think we should be helping Elizabeth to narrow down her choices of method to the most straightforward and familiar one that would help her answer her question.

Answer 14

I think it would be useful to ask Elizabeth which one she/he finds less intimidating.
Though heavy algebra is unavoidable in this exercise.

Answer 15

If you know Laplace expansion and determinants then I can tell another easy method.. (:

Answer 16

@xeno I don't know about Laplace expansion and determinants

Answer 17

then you must go with @Owlcoffee 's method

Answer 18

Thanks everyone. I got (-2,4.5) as my answer. Is that right?