10182

# 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.

### DETAILS

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, 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.
Reference
H. Anton and C. Rorres, Elementary Linear Algebra: Applications Version, New York: John Wiley & Sons, 1994.

### PERMANENT CITATION

(Instituto Superior Técnico)
 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.