Problem 80 - College Algebra

Question

Does \((x-3)^{2}+(y-5)^{2}=0\) represent the equation of a circle? If not, describe the graph of this equation.

Step-by-Step Solution

Answer

The equation \((x-3)^{2}+(y-5)^{2}=0\) represents a circle with its center at \((3, 5)\) and radius \(0\), which means it's actually a single point located at \((3, 5)\).

Step 1: Identify the coordinates of the center

Compare the given equation \((x-3)^{2}+(y-5)^{2}=0\) with the standard equation. The center of the circle can be obtained as \((h,k) = (3,5)\), where \(h\) and \(k\) represent the x and y coordinates of the center.

Step 2: Calculate the radius of the circle

Now observe the right side of the given equation, which should be \(r^{2}\). Since this is 0, the radius \(r = \sqrt{0} = 0\).

Step 3: Describe the graph of the equation

With these parameters, the circle has a center at \((3, 5)\) and a radius of \(0\). This means that the circle is just a point located at \((3, 5)\) in the plane.