Curves through Given Points in the Plane
In this Demonstration we construct curves through two, three, or five specified points in the plane. The three buttons correspond to a construction of a line through two distinct points, a circle through three distinct points, and a general conic section (ellipse, parabola, or hyperbola) through five distinct points, respectively. In each case, dragging the points, we can obtain degenerate curves by making one or more points coincide.
We can construct a line through two distinct points () and () in the plane by considering that both points satisfy the linear equation , that is, we should have and with coefficients , , and not all zero. Take these three equations together and we obtain a homogeneous linear system of three equations for the unknowns , , and . Since , , and are not all zero, the explicit equation for the given line can be written as .
In the case of a circle, three distinct points in a plane determine a curve with quadratic equation with coefficients , , , and not all zero. Once more, considering this equation together with the three equations for the distinct points (), () and (), we conclude the explicit equation for the circle as .
For a general conic section (ellipse, parabola, or hyperbola), five distinct points determine the curve uniquely. Using the same notation as in Conic Section (Wolfram MathWorld), the general quadratic equation has the form , with coefficients , , , , , and not all zero. Similarly to the previous cases of the line and the circle, the explicit equation for the given conic can be obtained by setting the determinant of the matrix corresponding to the homogeneous linear system of six equations for the unknowns , , , , , and , equal to zero.
H. Anton and C. Rorres, Elementary Linear Algebra: Applications Version, New York: John Wiley & Sons, 1994.