Problem 86 - 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. $$(0,0),(0,6),(3,3)$$

Step-by-Step Solution

Answer

The equation of the circle is \(x^{2}+y^{2}-3x-6y+0=0\)

Step 1: Substitute the given points into the equation

Substitute each of the points (0,0), (0,6), (3,3) into the equation \(x^{2}+y^{2}+D x+E y+F=0\). This will give the following three equations: Substituting (0,0) yields: \(0^{2} + 0^{2} + D(0) + E(0) + F = 0\) which simplifies to \(F = 0\)Substituting (0,6) yields: \(0^{2} + 6^{2} + D(0) + E(6) + F = 0\) which simplifies to \(E = -6\)Substituting (3,3) yields: \(3^{2} + 3^{2}+ D(3) + E(3) + F = 0\) which simplifies to \(D = -3\)

Step 2: Combine the results

Combine the results of the substitutions to form the equation of the circle. Using the coefficients D, E, F that were found, the equation is \(x^{2}+y^{2}-3x-6y+0=0\)

Step 3: Verification

To verify the solution, graph the equation \(x^{2}+y^{2}-3x-6y+0=0\) using a graphing utility or software. The circle should pass through the points (0,0), (0,6), (3,3).