Problem 43 - Algebra for College Students

Question

What is the graph of the equation \(x^{2}+y^{2}=0\) ? Explain your answer.

Step-by-Step Solution

Answer

The graph of the equation \(x^2 + y^2 = 0\) is a single point at the origin \((0, 0)\).

Step 1: Understand the Equation Form

The given equation is \(x^2 + y^2 = 0\). This is in the form of a circle equation \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle.

Step 2: Analyze the Only Possible Values

Since we know \(x^2\) and \(y^2\) are non-negative (i.e., \(x^2 \geq 0\) and \(y^2 \geq 0\)), the only way their sum can be zero is if both \(x^2 = 0\) and \(y^2 = 0\).

Step 3: Determine Values for Zero Equality

From \(x^2 = 0\), we get \(x = 0\) and from \(y^2 = 0\), we get \(y = 0\). Therefore, for the equation \(x^2 + y^2 = 0\) to hold true, \((x, y)\) must be \((0, 0)\).

Step 4: Interpret the Graph

The equation \(x^2 + y^2 = 0\) does not describe a circle with a positive radius since its center is at \((0, 0)\) and \(r^2 = 0\). This means it represents a single point at the origin.

Key Concepts

Circle Equation

In mathematics, a circle equation typically takes the form of \(x^2 + y^2 = r^2\), where \(x\) and \(y\) are the coordinates of any point on the circle, and \(r\) is the radius of the circle. This equation represents all the points in a plane that are a distance \(r\) from a fixed center point, usually denoted as \((h, k)\). For a standard circle equation centered at the origin, the equation simplifies to \(x^2 + y^2 = r^2\). This form is quite powerful, as it provides a simple way to check if any point \((x, y)\) lies on the circle. If it satisfies the equation, it is on the circle, otherwise it is not. The given exercise provides the equation \(x^2 + y^2 = 0\), which is a special case of the circle equation.

Radius

The radius is an essential component of a circle that measures the distance from the center of the circle to any point on its perimeter. In a circle equation \(x^2 + y^2 = r^2\), the radius \(r\) is derived from the term \(r^2\). Typically, the radius is a positive number, but in the given context, where the equation is \(x^2 + y^2 = 0\), the radius \(r\) is zero.
When the radius is zero, the circle collapses to a single point, rather than a shape with an area. This point is at the center of the circle, and here, since \(r = 0\), the center is the origin point \((0, 0)\). A radius of zero is unique in that the circle isn't visible as it lacks any size or circumference.

System of Equations

A system of equations is a collection of equations that are solved together. In the context of geometric shapes like circles, a system might include equations of multiple circles or other shapes that we need to evaluate at once.
Solving a system typically involves finding a set of values for the variables that satisfy all equations simultaneously. In the given exercise, however, there's effectively only one equation - \(x^2 + y^2 = 0\) - which behaves uniquely because it describes a degenerate circle, or point.
For this particular system, since both variables squared (\(x^2\) and \(y^2\)) must equal zero, each variable itself must be zero. Hence, \((x, y) = (0,0)\) is both the solution to this one-equation "system" and at the same time shows that there are no additional variables or factors, simplifying this to a unique result.

Origin Point

The origin point in a coordinate system is the point \((0,0)\), which acts as a central reference point for plotting on a graph. It is where the horizontal axis \(x\) and the vertical axis \(y\) intersect. This point is especially significant in the context of a circle equation when the radius equals zero.
In the equation \(x^2 + y^2 = 0\), both axes influence the position and dimension of the circle or point described by the equation. Given that neither \(x^2\) nor \(y^2\) can be negative, the only solution \(x = 0\) and \(y = 0\) implies that the graph of this equation is a single point at the origin.

• This point has no area or circumference as it is dimensionless.
• The origin serves as the center for conventional circles described by \(x^2 + y^2 = r^2\) when \(h = 0\) and \(k = 0\).

As such, origin-centered graphs form the basis for understanding more complex circles and transformations.