zmonk85 said:Homework Statement
I was looking at the following tutorial
http://mathworld.wolfram.com/Circle.html
Homework Equations
equations 31-34 o the link
The Attempt at a Solution
My question is just whether this means that for 31-34, the answers are determinants of 3x3 matricies?
Also, the nonliniarity for x^2 + y^2 is confusing. Do I treat it the same or do we have to get rid of the 2nd power by taking a derivative like in least squares?
Any help greatly appreciated.
The equation of a circle that passes through three points can be found using the circumcenter of the triangle formed by the three points. The circumcenter is the point where the perpendicular bisectors of the sides of the triangle intersect. Once the circumcenter is found, the equation of the circle can be written in the form (x - h)^2 + (y - k)^2 = r^2, where (h,k) is the coordinates of the circumcenter and r is the radius of the circle.
To find the circumcenter of a triangle, you can use the circumcenter formula, which is (x,y) = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3), where (x1,y1), (x2,y2), (x3,y3) are the coordinates of the three points that the circle passes through. Alternatively, you can also find the circumcenter by constructing the perpendicular bisectors of the sides of the triangle and finding their point of intersection.
No, a circle cannot pass through three points that are collinear (lie on the same line). This is because a circle is defined by a center point and a radius, and for a circle to pass through three points, the center point must be equidistant from all three points. However, if the points are collinear, there is no single center point that is equidistant from all three points.
Yes, there are two special cases when finding the equation of a circle passing through three points: when the three points are collinear (as mentioned above), and when the three points are the vertices of an equilateral triangle. In the case of an equilateral triangle, the circumcenter will be the centroid (center of mass) of the triangle, and the radius of the circle will be equal to one-third of the length of any side of the triangle.
Yes, you can use the equation of a circle to find its area and circumference. The area of a circle can be found using the formula A = πr^2, where r is the radius of the circle. The circumference can be found using the formula C = 2πr, where r is again the radius of the circle. Once you have found the equation of the circle that passes through three points, you can simply plug in the value of r to find the area and circumference.