![\[\begin{array}{*{35}{l}} \left( \mathbf{i} \right)\text{ }{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }{{\mathbf{y}}^{\mathbf{2}}}~+\text{ }\mathbf{6x}\text{ }\text{ }-\mathbf{8y}\text{ }-\text{ }\mathbf{24}\text{ }=\text{ }\mathbf{0} \\ \left( \mathbf{ii} \right)\text{ }\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{2}{{\mathbf{y}}^{\mathbf{2}}}~-\text{ }\mathbf{3x}\text{ }+\text{ }\mathbf{5y}\text{ }=\text{ }\mathbf{7} \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c42bc953bc55b4dd30fb052b0f2bf7ca_l3.png "Rendered by QuickLaTeX.com")
(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 =
![\[\begin{array}{*{35}{l}} \left( -6/2,\text{ }-\left( -8 \right)/2 \right) \\ =\text{ }\left( -3,\text{ }4 \right) \\ Radius\text{ }=~\surd \left( {{a}^{2}}~+\text{ }{{b}^{2}}~\text{ }c \right) \\ =~\surd \left( {{3}^{2}}~+\text{ }{{4}^{2}}~\text{ }-\left( -24 \right) \right) \\ =~\surd \left( 9\text{ }+\text{ }16\text{ }+\text{ }24 \right) \\ =~\surd \left( 49 \right) \\ =\text{ }7 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-fb1a59f91449440aabf9fe024aa6b2b9_l3.png "Rendered by QuickLaTeX.com")
∴ 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)
