Chapter 11, Problem 100RE

To determine

Find the general equation of the circle.

Answer to Problem 100RE

  $\boxed{x^2+y^2+2x+2y-3=0}$

Explanation of Solution

Given information:

Find the general equation of the circle that passes through the three points $\left(0,1\right),\left(1,0\right)$ and $\left(-2,1\right)$

[hint: the general equation of the circle is $x^2+y^2+Dx+Ey+F=0$ ]

Calculation:

Consider the general equation of the circle is $x^2+y^2+Dx+Ey+F=0$ passes through the three points $\left(0,1\right),\left(1,0\right)$ and $\left(-2,1\right)$

These point must satisfy the given relation since they lie on the given circle.

  $\begin{array}{l}{\left(0\right)}^2+{\left(1\right)}^2+D\times 0+E\times 1+F=0\\ E+F+1=\mathrm{0.......}\left(1\right)\\ {\left(1\right)}^2+{\left(0\right)}^2+D\times 1+E\times 0+F=0\\ D+F+1=\mathrm{0........}\left(2\right)\\ {\left(-2\right)}^2+{\left(1\right)}^2+D\times -2+E\times 1+F=0\\ -2D+E+F+5=\mathrm{0.......}\left(3\right)\end{array}$

Now solve the value of variable by using augmented matrix,

  $A=\left[\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ -2& 1& 1\end{array}\right],X=\left[\begin{array}{c}D\\ E\\ F\end{array}\right]andB\left[\begin{array}{c}-1\\ -1\\ -5\end{array}\right]$

Now solve for $AX=B$ , in order to determine the unknown variable means matrix $X$ so multiply above equation by $A^{-1}$ we get,

  $\begin{array}{l}{A}^{-1}\left(AX\right)=A^{-1}B\\ \left(A^{-1}A\right)=A^{-1}B\\ I_{3}X=A^{-1}B\\ X=A^{-1}B\end{array}$

Now determine $A^{-1}$ by using graphing utility.

  $A^{-1}=\left[\begin{array}{ccc}\frac{1}{2}& 0& -\frac{1}{2}\\ \frac{3}{2}& -1& -\frac{1}{2}\\ -\frac{1}{2}& 1& \frac{1}{2}\end{array}\right]$ now product of matrix $A^{-1}B$ will be,

  $\begin{array}{l}X=\left[\begin{array}{ccc}\frac{1}{2}& 0& -\frac{1}{2}\\ \frac{3}{2}& -1& -\frac{1}{2}\\ -\frac{1}{2}& 1& \frac{1}{2}\end{array}\right]\times \left[\begin{array}{c}-1\\ -1\\ -5\end{array}\right]\\ =\left[\begin{array}{ccc}\left(-1\right)\frac{1}{2}& +\left(-1\right)\times 0& +\left(-5\right)\times \left(-\frac{1}{2}\right)\\ \left(-1\right)\times \frac{3}{2}& +\left(-1\right)\times -1& +\left(-5\right)\times \left(-\frac{1}{2}\right)\\ \left(-1\right)\times -\frac{1}{2}& +\left(-1\right)\times 1& +\left(-5\right)\times \frac{1}{2}\end{array}\right]\\ =\left[\begin{array}{ccc}-\frac{1}{2}& +0& +\frac{5}{2}\\ -\frac{3}{2}& +1& +\frac{5}{2}\\ \frac{1}{2}& -1& -\frac{5}{2}\end{array}\right]\\ X=\left[\begin{array}{c}2\\ 2\\ -3\end{array}\right]\end{array}$

The value of variables $D=2,E=2andF=-3$

Hence the equation of the circle is $\boxed{x^2+y^2+2x+2y-3=0}$ .