Problem 88 - 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. $$(0,0),(0,-2),(3,0)$$

Step-by-Step Solution

Answer

The equation of the circle is \(x^{2}+y^{2}+3x-2y=0\).

Step 1: Substituting points into the equation

Substitute the points (0,0), (0,-2), and (3,0) into the equation, to get three equations. The first point (0,0) gives \(0=0+D*0+E*0+F\), which simplifies to \(F=0\). The second point (0,-2) gives \(-2^{2}=4+D*0+E*(-2)+0\), which simplifies to \(4=-2E\). The third point (3,0) gives \(3^{2}=9+D*3+E*0+0\), which simplifies to \(9=3D\).

Step 2: Solve the equations for D and E

Solving the equation \(4=-2E\) gives \(E=-2\). The equation \(9=3D\) gives \(D=3\).

Step 3: Write the equation of the circle

Substitute \(D=3\), \(E=-2\), and \(F=0\) back into the general equation to get \(x^{2}+y^{2}+3x-2y=0\) as the equation for the circle.

Step 4: Verification

Graph the circle using the equation \(x^{2}+y^{2}+3x-2y=0\) and plot the points. Make sure the points are on the circumference of the circle.