The equation of the line:	c1 x + c2 y + c3 = 0
The equation for the first point:	c1 x1 + c2 y1 + c3 = 0
The equation for the second point:	c1 x2 + c2 y2 + c3 = 0

The three equations above form a linear system of equations, which we write in the matrix form:

x y 1 x1 y1 1 x2 y2 1

c1 c2 c3

= 0 0 0

From this we obtain:

A. The equation of a line passing through the two distinct points (x₁ , y₁ ) and (x₂ , y₂ ) is

	x	y	1		= 0.
	x1	y1	1
	x2	y2	1

Example 1: The equation of a line

B. The equation of a circle passing through the three distinct points (x₁ , y₁ ), (x₂ , y₂ ) and (x₃ , y₃ ) is

	x2 + y2	x	y	1		= 0.
	x21 + y21	x1	y1	1
	x22 + y22	x2	y2	1
	x23 + y23	x3	y3	1

Example 2: The equation of a circle

C. The equation of a conic section(parabola, hyperbola or an ellipse) passing through the five distinct points (x_i , y_i ),  i = 1, ..., 5 is

	x2	xy	y2	x	y	1		= 0.
	x21	x1 y1	y21	x1	y1	1
	x22	x2 y2	y22	x2	y2	1
	x23	x3 y3	y23	x3	y3	1
	x24	x4 y4	y24	x4	y4	1
	x25	x5 y5	y25	x5	y5	1

D. The equation of a plane passing through the three distinct points (x_i , y_i , z_i ),  i = 1, 2, 3, which are not collinear, i.e., they do not lie on a same line, is

	x	y	z	1		= 0.
	x1	y1	z1	1
	x2	y2	z2	1
	x3	y3	z3	1

E. The equation of a spherical surface passing through the four distinct points (x_i , y_i , z_i ),  i = 1, ..., 4, which are not coplanar, i.e., they do not lie on a same plane, is

	x2 + y2 + z2	x	y	z	1		= 0.
	x21 + y21 + z21	x1	y1	z1	1
	x22 + y22 + z22	x2	y2	z2	1
	x23 + y23 + z23	x3	y3	z3	1
	x24 + y24 + z24	x4	y4	z4	1