Problem 66 - 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 a radius of zero or in other words, it is a point located at (3,5) on a graph.

Step 1: Identify the Equation

The given equation can be written in the form of \( (x - h)^2 + (y - k)^2 = r^2 \). Here, (h,k) represents the center of a circle, and r represents the radius of the circle. In the given equation, \( h = 3 \), \( k = 5 \), and \( r = 0 \).

Step 2: Determine the Implication of a Radius of Zero

A circle usually has a radius greater than zero. In our case, having a radius of zero specifies that all the points of the circle collapse to a single point at the center. In this case, (3,5). A circle with radius zero is usually called a point or a zero circle.

Step 3: Describe the Graph of the Equation

The graph of the equation \((x - 3)^2 + (y - 5)^2 = 0\) is just a point located at (3,5) as the radius is zero.