If the equation ${\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{hxy}+2\mathrm{gx}+2\mathrm{fy}+\mathrm{c}=0$ represents a circle, then the condition for that circle to pass through three quadrants only but not passing through the origin is

$$\begin{gathered} \text{ The given circle is }{\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{hxy}+2\mathrm{gx}+2\mathrm{fy}+\mathrm{c}=0 -----\left(\mathrm{i}\right) \\ \mathrm{For} \left(\mathrm{i}\right) \mathrm{to} \mathrm{represent} \mathrm{a} \mathrm{circle} , \mathrm{h} = 0. \\ \mathrm{So}, \mathrm{the} \mathrm{given} \mathrm{circle} \mathrm{is} \\ {\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{gx}+2\mathrm{fy}+\mathrm{c}=0 ------\left(\mathrm{ii}\right) \end{gathered}$$
 

For circle (ii) to pass through three quadrants only,
I. Circle must cut x-axis. i.e., ${\mathrm{g}}^2-\mathrm{c}>0$.
II. Circle must cut y-axis. g² - c> 0.
III. The origin should lie outside circle (ii), i.e., c > 0.
Therefore, the required conditions are ${\mathrm{g}}^2>\mathrm{c},{\mathrm{f}}^2>\mathrm{c},\mathrm{c}>0$ and h = 0.