Question

Equation of a circle when 3 points are given using matrices

Answer 1

Answer:

A circle has an equation of the form

x² + y² - ax - by - c = 0

Putting in values for x and y gives a linear equation for a, b, c.

Given three points (x₁, y₁), (x₂, y₂) and (x₃, y₃), we get a system of three linear equations for a, b, c that we can represent (and solve) using matrices:

x₁a + y₁b + c = x₁² + y₁²

x₂a + y₂b + c = x₂² + y₂²

x₃a + y₃b + c = x₃² + y₃²

In matrix form:

$\displaystyle\left(\begin{array}{ccc}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{array}\right)\left(\begin{array}{c}a\\b\\c\end{array}\right) = \left(\begin{array}{c}x_1^2+y_1^2\\x_2^2+y_2^2\\x_3^2+y_3^2\end{array}\right)$

How you then solve the system is up to you (or what the exercise says).