Equation of a Circle Through Two Points and a Line Passing Through its Center

Consider the general equation a circle is given by
\[{x^2} + {y^2} + 2gx + 2fy + c = 0\]

If the given circle is passing through two points, say $$A\left( {{x_1},{y_1}} \right)$$and $$B\left( {{x_2},{y_2}} \right)$$, then these points must satisfy the general equation of a circle. Now put these two points in the given equation of a circle, i.e.:

\[\begin{gathered} {x_1}^2 + {y_1}^2 + 2g{x_1} + 2f{y_1} + c = 0\,\,\,{\text{ – – – }}\,\left( {\text{i}} \right) \\ {x_2}^2 + {y_2}^2 + 2g{x_2} + 2f{y_2} + c = 0\,\,\,{\text{ – – – }}\,\left( {{\text{ii}}} \right) \\ \end{gathered} \]

Also, the given straight line $$ax + by + {c_1} = 0$$ passes through the center $$\left( { – g, – f} \right)$$ of the circle.

\[\begin{gathered} \Rightarrow a\left( { – g} \right) + b\left( { – f} \right) + {c_1} = 0\, \\ \Rightarrow \, – ag – bf + {c_1} = 0\,\,\,{\text{ – – – }}\,\,\,\left( {{\text{iii}}} \right) \\ \end{gathered} \]

To evaluate the equation of the required circle, we must the find the values of $$g,f,c$$ from the above equations (i), (ii) and (iii) and put these in the first equation of a circle. We can solve these three using the method of simultaneous equations.

Example: Find the equation of a circle through two points$$\left( {1,2} \right)$$,$$\left( {2,3} \right)$$ and whose center is on the straight line $$x – y + 1 = 0$$.

Solution: Consider the required equation of a circle in general form as

\[{x^2} + {y^2} + 2gx + 2fy + c = 0\,\,\,{\text{ – – – }}\left( {\text{A}} \right)\]

We know that the given points $$\left( {1,2} \right)$$, $$\left( {2,3} \right)$$ lie on the circle, and if we put these points in the above equation of a circle, (A) becomes for these three points:

\[\begin{gathered} 2g + 4f + c = – 5\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ 4g + 6f + c = – 13\,\,\,{\text{ – – – }}\left( {{\text{ii}}} \right) \\ \end{gathered} \]

Since the center of the circle is $$\left( { – g, – f} \right)$$ and this center lies on the given straight line, $$\left( { – g, – f} \right)$$ must satisfy the equation of a line as

\[ – g + f + 1 = 0\,\,\,{\text{ – – – }}\left( {{\text{iii}}} \right)\]

Now we solve equations (i), (ii) and (ii), using the method of simultaneous equations and we use the values of $$f = – \frac{5}{2}$$ ,$$g = – \frac{3}{2}$$ and $$c = 8$$

Now putting these three values in the first equation (A), we get the required equation of a circle passing through two points and with its center lying on the line.

\[{x^2} + {y^2} – 3x – 5y + 8 = 0\]

This is the required equation of a circle.