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 x2+y2+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)tofindf=−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.