Find the equation of the circle passing through points (1,2), (3,4), and (5,2).

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Step-by-Step Solution

Step 1: General Equation of a Circle

The general equation of a circle is given by x^2 + y^2 + 2gx + 2fy + c = 0$. We need to find the values of g$, f$, and c using the given points.

Step 2: Substitute the points into the equation

We substitute each of the given points (1,2)$, (3,4)$, and (5,2) into the general equation of the circle. This gives us a system of three linear equations with three variables g$, f$, and c$.

Step 3: Solve the system of equations

To solve the system, we subtract equation (1) from equation (2) and equation (1) from equation (3). This eliminates c and simplifies the system to two equations with g and f$. From the second subtraction, we find g = -3$.

Step 4: Find f and c

Now that we have g = -3$, we substitute this value into equation (4) to find f = -2$. Finally, we substitute g = -3 and f = -2 into equation (1) to find c = 9$.

Step 5: Write the equation of the circle

Substitute the values of g = -3$, f = -2$, and c = 9 back into the general equation of the circle to obtain the final equation.