Problem 68 - Algebra 2

Question

a. Writing Explain why \(x^{2}+y^{2}=0\) does not represent a circle. b. Critical Thinking What does the equation represent?

Step-by-Step Solution

Answer

a. This equation does not represent a circle, because the value of the radius equals zero, suggesting a point rather than a circle. b. The given equation represents a single point located at the origin, i.e., (0,0).

Step 1: Understand why isn't it a circle

The equation of a circle is given by \(x^{2}+y^{2}=r^{2}\), where \(r\) is the radius of the circle. In this case, when comparing this equation with given \(x^{2}+y^{2}=0\), it can be said that \(r^{2}=0\), therefore, the radius \(r\) would be zero. A circle with a radius of zero is essentially a point. It doesn't share the properties of a circle, because it has no circumference, so it wouldn't be right to term it as a circle.

Step 2: Understand what the equation represents

Since the given equation is of the form \(x^{2}+y^{2}=r^{2}\) and the \(r\) in this case equals to zero, the equation represents a single point at the origin of the coordinates, which is (0,0). A point at the origin is the only case which can satisfy this equation, as putting any other value for \(x\) and \(y\) would violate the given equation.

Key Concepts

Understanding Radius

The concept of radius is fundamental in the study of circles. The radius of a circle is the distance from its center to any point on its boundary. It is represented by the letter \(r\) in the circle's equation \(x^2 + y^2 = r^2\). This equation tells us that all points \((x, y)\) lying on the circle are exactly \(r\) units away from the center of the circle at the origin.
The radius is always a non-negative number. A circle with a positive radius forms a round shape with a circumference. When the radius becomes zero, the circle compresses to a single point.

• In geometric terms: The radius is half of the diameter.
• In the equation \(x^2 + y^2 = r^2\), \(r\) defines the stretch of the circle in all directions.
• A zero radius implies no area, reducing the circle to just a point.

The Significance of the Origin

The origin is a key concept in any coordinate system, particularly when discussing circles. It is the starting point of the coordinate system and is located where the \(x\)-axis and \(y\)-axis intersect, i.e., at (0,0).
When we talk about a circle on a coordinate plane, its equation \(x^2 + y^2 = r^2\) usually assumes the circle is centered at the origin, unless otherwise specified. This simplifies our calculations and understanding.

• The origin is crucial because it is often the center of circles in standard equations.
• A circle centered at the origin with a zero radius results in a single point at the origin itself.
• The origin provides a fixed reference point for measuring distances and angles.

Understanding the Coordinate System

A coordinate system, especially the Cartesian coordinate system, plays an essential role in geometry. It helps in identifying each point in a plane by a pair of numerical coordinates,\((x, y)\).
In this system, the equation of a circle \(x^2 + y^2 = r^2\) utilizes these coordinates to define a set of all points \((x, y)\) that are equidistant from a fixed center, usually the origin.

• The coordinate system is a grid that allows precise plotting of circles and other shapes.
• Each point's coordinates describe its position relative to the origin.
• It provides a universal framework for expressing geometric shapes mathematically.

Using a coordinate system enhances our ability to solve mathematical problems and visualize abstract concepts like circles.