Problem 85 - 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. $$(0,0),(5,5),(10,0)$$

Step-by-Step Solution

Answer

The equation of the circle that passes through the points (0,0), (5,5), (10,0) is \(x^{2}+y^{2}-10x-10y=0\).

Step 1: Substitution of Points into the General Equation of a Circle

Substitute the provided points (0,0),(5,5), (10,0) into the given general equation of a circle. Here we obtain three equations: \n1) From (0,0): \(F = 0\). 2) From (5,5): \(25+25+5D+5E+F=0\). 3) From (10,0): \(100+0+10D+0+F=0\).

Step 2: Solve the system of equations

Now, solve this system for variables D,E and F. As we know from step 1, \(F=0\). Substitute \(F=0\) into the equation from point (10,0) to obtain \(D = -10\). Finally, substitute \(F=0\) and \(D=-10\) into the equation from point (5,5) to get \(E=-10\).

Step 3: Construct the Equation of the Circle

Substitute D,E and F into the equation \(x^{2}+y^{2}+D x+E y+F=0\), your final equation becomes \(x^{2}+y^{2}-10x-10y=0\).

Step 4: Verification

To verify the result, graph the circle using the equation found and mark the points (0,0),(5,5), (10,0). If all three points lie on the circle, then the equation is correct.