(iii) equation of the circle: x2 + y2 + 2ax + 2by + c = 0 ….. (1)
Substituting the point (5, -8) in equation (1), we get
\[\begin{array}{*{35}{l}}
{{5}^{2}}~+\text{ }{{\left( -\text{ }8 \right)}^{2}}~+\text{ }2a\left( 5 \right)\text{ }+\text{ }2b\left( -\text{ }8 \right)\text{ }+\text{ }c\text{ }=\text{ }0 \\
25\text{ }+\text{ }64\text{ }+\text{ }10a\text{ }-\text{ }16b\text{ }+\text{ }c\text{ }=\text{ }0 \\
10a\text{ }-\text{ }16b\text{ }+\text{ }c\text{ }+\text{ }89\text{ }=\text{ }0\ldots ..\text{ }\left( 2 \right) \\
\end{array}\]
Substituting the points (-2, 9) in equation (1), we get
\[\begin{array}{*{35}{l}}
{{\left( -\text{ }2 \right)}^{2}}~+\text{ }{{9}^{2}}~+\text{ }2a\left( -\text{ }2 \right)\text{ }+\text{ }2b\left( 9 \right)\text{ }+\text{ }c\text{ }=\text{ }0 \\
4\text{ }+\text{ }81\text{ }\text{ }4a\text{ }+\text{ }18b\text{ }+\text{ }c\text{ }=\text{ }0 \\
-4a\text{ }+\text{ }18b\text{ }+\text{ }c\text{ }+\text{ }85\text{ }=\text{ }0\ldots ..\text{ }\left( 3 \right) \\
\end{array}\]
Substituting the points (2, 1) in equation (1), we get
\[\begin{array}{*{35}{l}}
{{2}^{2}}~+\text{ }{{1}^{2}}~+\text{ }2a\left( 2 \right)\text{ }+\text{ }2b\left( 1 \right)\text{ }+\text{ }c\text{ }=\text{ }0 \\
4\text{ }+\text{ }1\text{ }+\text{ }4a\text{ }+\text{ }2b\text{ }+\text{ }c\text{ }=\text{ }0 \\
4a\text{ }+\text{ }2b\text{ }+\text{ }c\text{ }+\text{ }5\text{ }=\text{ }0\ldots ..\text{ }\left( 4 \right) \\
\end{array}\]
By simplifying equations (2), (3), (4) we get
a = 58, b = 24, c = – 285.
by substituting the values of a, b, c in equation (1), we get
\[\begin{array}{*{35}{l}}
{{x}^{2}}~+\text{ }{{y}^{2}}~+\text{ }2\left( 58 \right)x\text{ }+\text{ }2\left( 24 \right)\text{ }-\text{ }285\text{ }=\text{ }0 \\
{{x}^{2}}~+\text{ }{{y}^{2}}~+\text{ }116x\text{ }+\text{ }48y\text{ }-\text{ }285\text{ }=\text{ }0 \\
\end{array}\]
∴ The equation of the circle is x2 + y2 + 116x + 48y – 285 = 0
(iv) the equation of the circle: x2 + y2 + 2ax + 2by + c = 0 ….. (1)
Substituting the points (0, 0) in equation (1), we get
\[\begin{array}{*{35}{l}}
{{0}^{2}}~+\text{ }{{0}^{2}}~+\text{ }2a\left( 0 \right)\text{ }+\text{ }2b\left( 0 \right)\text{ }+\text{ }c\text{ }=\text{ }0 \\
0\text{ }+\text{ }0\text{ }+\text{ }0a\text{ }+\text{ }0b\text{ }+\text{ }c\text{ }=\text{ }0 \\
c\text{ }=\text{ }0\ldots ..\text{ }\left( 2 \right) \\
\end{array}\]
Substituting the points (-2, 1) in equation (1), we get
\[\begin{array}{*{35}{l}}
{{\left( -\text{ }2 \right)}^{2}}~+\text{ }{{1}^{2}}~+\text{ }2a\left( -\text{ }2 \right)\text{ }+\text{ }2b\left( 1 \right)\text{ }+\text{ }c\text{ }=\text{ }0 \\
4\text{ }+\text{ }1\text{ }\text{ }4a\text{ }+\text{ }2b\text{ }+\text{ }c\text{ }=\text{ }0 \\
-4a\text{ }+\text{ }2b\text{ }+\text{ }c\text{ }+\text{ }5\text{ }=\text{ }0\ldots ..\text{ }\left( 3 \right) \\
\end{array}\]
Substitute the points (-3, 2) in equation (1), we get
\[\begin{array}{*{35}{l}}
{{\left( -\text{ }3 \right)}^{2}}~+\text{ }{{2}^{2}}~+\text{ }2a\left( -\text{ }3 \right)\text{ }+\text{ }2b\left( 2 \right)\text{ }+\text{ }c\text{ }=\text{ }0 \\
9\text{ }+\text{ }4\text{ }-\text{ }6a\text{ }+\text{ }4b\text{ }+\text{ }c\text{ }=\text{ }0 \\
-6a\text{ }+\text{ }4b\text{ }+\text{ }c\text{ }+\text{ }13\text{ }=\text{ }0\ldots ..\text{ }\left( 4 \right) \\
\end{array}\]
By simplifying the equations (2), (3), (4) we get
a = -3/2, b = -11/2, c = 0
by substituting the values of a, b, c in equation (1), we get
\[\begin{array}{*{35}{l}}
{{x}^{2}}~+\text{ }{{y}^{2}}~+\text{ }2\left( -3/2 \right)x\text{ }+\text{ }2\left( -11/2 \right)y\text{ }+\text{ }0\text{ }=\text{ }0 \\
{{x}^{2}}~+\text{ }{{y}^{2}}~-\text{ }3x\text{ }-\text{ }11y\text{ }=\text{ }0 \\
\end{array}\]
∴ The equation of the circle is x2 + y2 – 3x – 11y = 0