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Chapter 2: Problem 66

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

Short Answer

Expert verified
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 by step solution

01

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 \( r = 0 \).
02

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.
03

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.

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