Equation of a circle passing through 3 points (x1, y1) (x2, y2) and (x3, y3).
The equation of the circle is described by the equation:
After substituting the three given points which lies on the circle we get the set of equations that can be described by the determinant:
The coefficienta A, B, C and D can be found by solving the following determinants:
The values of A, B, C and D will be after solving the determinants:
Center point (x, y) and the radius of a circle passing through 3 points (x1, y1) (x2, y2) and (x3, y3) are:
Example: Find the equation of a circle passing through the points (⎯ 3, 4), (4, 5) and (1, ⎯ 4).
A = ⎯ 3(5 ⧾ 4) ⎯ 4(4 ⎯ 1) ⧾ 4(⎯ 4) ⎯ 1 • 5 = ⎯ 60
B = (9 ⧾ 16)(⎯ 4 ⎯ 5) ⧾ (16 ⧾ 25)(4 ⧾ 4) ⧾ (1 ⧾ 16)(5 ⎯ 4) = 120
C = (9 ⧾ 16)(4 ⎯ 1) ⧾ (16 ⧾ 25)(1 ⧾ 3) ⧾ (1 ⧾ 16)(⎯ 3 ⎯ 4) = 120
D = (9 ⧾ 16)(1 • 5 ⎯ 4(⎯ 4)) ⧾ (16 ⧾ 25)(⎯ 3 • (⎯ 4) ⎯ 1 · 4) ⧾ (1 ⧾ 16)(4 • 4 ⎯ (⎯ 3)5) = 1380
Divide all terms by ⎯ 60 to obtaine:
The center of the circle is by solving x and y is at point (1, 1)
The radius of the circle is:
The
equation of the circle represented by standard form is: