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.

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

Answer: The equation of the circle is $x^2 + y^2 - 6x - 4y + 9 = 0$ .

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.

More from this chapter

All Conic Sections solutions →

Solve your own maths question free →