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

Consider the general equation a circle is given by

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.:

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

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

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:

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

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.

This is the required equation of a circle.