(i) The equation of the circle is x2 + y2 + 6x – 8y – 24 = 0 …… (1)
Since, for a circle x2 + y2 + 2ax + 2by + c = 0 …… (2)
Centre = (-a, -b)
So by comparing equation (1) and (2)
Centre =
\[\begin{array}{*{35}{l}}
\left( -6/2,\text{ }-\left( -8 \right)/2 \right) \\
=\text{ }\left( -3,\text{ }4 \right) \\
Radius\text{ }=~\surd \left( {{a}^{2}}~+\text{ }{{b}^{2}}~\text{ }c \right) \\
=~\surd \left( {{3}^{2}}~+\text{ }{{4}^{2}}~\text{ }-\left( -24 \right) \right) \\
=~\surd \left( 9\text{ }+\text{ }16\text{ }+\text{ }24 \right) \\
=~\surd \left( 49 \right) \\
=\text{ }7 \\
\end{array}\]
∴ The centre of the circle is (-3, 4) and the radius is 7.
(ii) The equation of the circle is 2x2 + 2y2 – 3x + 5y = 7 (divide by 2 we get)
x2 + y2 – 3x/2 + 5y/2 = 7/2
comparing with the equation x2 + y2 + 2ax + 2by + c = 0
Centre = (-a, -b)