Does x^2 y^2 25 Represent a Point?

When first encountering the equation x^2 y^2 25, the question may arise whether it represents a single point or an entire geometric figure. The short answer is that it represents a circle, not a point. Let's delve into the mathematics and graphing that supports this conclusion.

The Equation of a Circle

The standard form of a circle's equation is:

(x - a)^2 (y - b)^2 r^2

Here, the circle is centered at the point (a, b), and the radius is r. In the given equation x^2 y^2 25, we can see: Center a 0 and b 0 (the origin of the Cartesian plane) Radius r^2 25, so r 5

Graphing the Equation

The equation x^2 y^2 25 can be rewritten as:

y^2 25 - x^2

From this, we can find the general form of y:

y ±√(25 - x^2)

The graph of this equation is a circle with its center at the origin (0, 0) and a radius of 5. This means that all points (x, y) that satisfy the equation are exactly 5 units away from the origin. Here’s how it looks:

Verifying the Properties of a Point

A single point in the Cartesian plane is represented as (x, y) where the distance from the origin to this point is 0. In the equation x^2 y^2 25, the radius is 5, which is not 0. Therefore, this equation cannot represent a single point.

Instead, it represents all the points that are exactly 5 units away from the origin, forming a circle. This is consistent with the properties of a circle and the given equation x^2 y^2 25.

To reiterate:

The equation x^2 y^2 25 represents a circle centered at (0, 0) with a radius of 5 units. A single point in the Cartesian plane would have a radius of 0, but x^2 y^2 25 has a non-zero radius of 5.

This clear-cut mathematical representation and verification answer the question definitively. The equation indeed represents a circle, not a single point.