Problem 87 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the equation of the circle $$x^{2}+y^{2}+D x+E y+F=0$$ that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle. $$(-3,-1),(2,4),(-6,8)$$

Step-by-Step Solution

Answer

The exact answer depends on the solution to the system of equations, which varies with D, E, F. The equation for the circle would take the form \(x^{2} +y^{2} + D*x + E*y + F = 0\), with appropriate values substituted for D, E, and F. Verification will be done by plotting the derived equation and points in a graphing utility.

Step 1: Set up the system of equations

Apply each of the points to the general equation of the circle: \(-3^{2} + -1^{2} + D*-3 + E*-1 + F = 0\), \(2^{2} + 4^{2} + D*2 + E*4 + F = 0\), and \(-6^{2} + 8^{2} + D*-6 + E*8 + F = 0\). This results in the system of equations: \(9 + 1 - 3D - E + F = 0\), \(4 + 16 + 2D + 4E + F = 0\), and \(36 + 64 - 6D +8E + F = 0\).

Step 2: Solve the system of equations

Solving this system of equations will give the values of D, E, and F. One common method is to reduce this system to two equations in two unknowns by combinations, then solve for one variable in one of the resulting equations and substitute this expression in the other to solve for the other variable.

Step 3: Substitute back into the circle equation

Once D, E and F are found, substitute these values back into the general equation of the circle \(x^{2} +y^{2} + D*x + E*y + F = 0\), to get the specific equation of the circle that passes through the given points.

Step 4: Graph the circle and points

Use a graphing utility to plot the three points and graph the circle from the derived equation. This will visually confirm the derived equation is indeed the correct equation of the circle passing through the given points.