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


two-line-equation-circle

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.