Question

Given three points, (-1,3.2), (-8,4), and (-6.5,-9.3), determine the equation of the circle that passes through the points. solve the problem by deriving a system of three linear equations (substi- tute the points in the equation) and solve the system.

Answer 1

The equation of the circle can be written in general form as ...

... x² +y² +ax +by +c = 0

Substituting the given points, this equation becomes

... (-1)² +3.2² + (-1)a +3.2b +c = 0

... (-8)² +4² + (-8)a +4b +c = 0

... (-6.5)² +(-9.3)² + (-6.5)a +(-9.3)b +c = 0

In augmented matrix form, the system of equations is

$\left[\begin{array}{ccc|c}-1&3.2&1&-11.24\\-8&4&1&-80\\-6.5&-9.3&1&-128.74\end{array}\right]$

The solution to this set of equations can be found using the matrix solver on your calculator. It is ...

$a\approx 10.3754080522\\b\approx 4.83482045702\\c\approx -16.3360174102$

So, the equation for the circle in general form is

... x² + y² + 10.3754x + 4.8348y - 16.336 = 0