Answer to Question #129861 in Analytic Geometry for Grace kimanzi

Question

A circle passes through the points A(-1,2),B (3,4) C(2,3).determine it's equation

Answer 1

$$Solution$$

Start with the General Form of circle.

Substitution of A, B, C.

Let the circle be $x^2+y^2+Dx+Ey+F=0$

$(-1,2),\ (-1)^2+2^2-D+2E+F=0\\ (3,4),\ 3^2+4^2+3D+4E+F=0\\ (2,3),\ 2^2+3^2+2D+3E+F=0$

Simplification

$D-2E-F=5...........(1)\\ 3D+4E+F=-25.....(2)\\ 2D+3E+F=-13.....(3)$

Solve the equations.

$2(1)+(2),\ D=-3\\ (2)+(3),\ D+E=-12\\ (-3)+E=-12 \implies E=-12+3=-9\\ (-3)-2(-9)-F=5\\ \implies 15-F=5\\ \implies -F=-10 \implies F=10$

Substitute D, E, F in the General Form.

The equation of the circle is $x^2+y^2-3x-9y+10=0$