Equation of Circle through Two Points and Line Passes through its Center

Consider the general equation 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 circle. Now put these two points in the given equation of 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 centre \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 required circle, we must the find values of g,f,c from the above equation (i), (ii) and (iii) and put these in the first equation of circle. We can solve these three using the method of simultaneous equations.

Example: Find the equation of circle through two points\left(  {1,2} \right),\left( {2,3} \right) and whose centre is on the straight linex - y  + 1 = 0.
Solution: Consider the required equation of 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) lies on the circle and now put these points in the above equation of 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 centre of the circle is \left( { - g, - f} \right) and this centre lies on the given straight line, so that \left(  { - g, - f} \right) must satisfy the equation of line as

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


Now solving the equations (i), (ii) and (ii), using method of simultaneous equation and we the values of f =  -  \frac{5}{2} ,g =  - \frac{3}{2} and c = 8
Now putting all these three values in the first equation (A), to get the required equation of circle passing through two points and its centre lies on the line.

{x^2}  + {y^2} - 3x - 5y + 8 = 0


Which is the required equation of circle.

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