Problem 16 - Calculus Single Variable

Question

A circle is described in words. Give its Cartesian equation. The circle with center at the origin and radius 2

Step-by-Step Solution

Answer

The Cartesian equation is \\(x^2 + y^2 = 4\\).

Step 1: Understand the General Equation of a Circle

The general equation of a circle with center \(h, k\) and radius \ r \ is given by: \((x - h)^2 + (y - k)^2 = r^2\). This equation represents all the points \(x, y\) that are equidistant \(r\) from the center \(h, k\).

Step 2: Identify Given Values

In this problem, the circle's center is at the origin \(0, 0\) and the radius is 2. Thus, \(h = 0\), \(k = 0\), and \(r = 2\).

Step 3: Substitute Values into General Equation

Substitute the values \(h = 0\), \(k = 0\), and \(r = 2\) into the general equation: \((x - 0)^2 + (y - 0)^2 = 2^2\). This simplifies to: \(x^2 + y^2 = 4\).

Key Concepts

General Equation of a Circle

In mathematics, the Cartesian equation of a circle is foundational for understanding how circles are represented in a plane. A circle in a Cartesian coordinate system can be described with the general equation:

• \((x - h)^2 + (y - k)^2 = r^2\)

Here, \( (h, k) \) represents the center of the circle and \( r \) is the radius. This equation tells us that the set of all points \((x, y)\) that satisfy the equation are exactly \( r \) units away from \((h, k)\).
It's important because it allows us to visualize and understand the position and distance properties of a circle relative to a coordinate plane.
By simply substituting the values for the center and radius into this equation, we can model any circle accurately. The beauty of this equation is its simplicity—once understood, it becomes a powerful tool for solving many geometrical problems related to circles.

Center of a Circle

The circle's center is a crucial part of its definition in the coordinate plane. In the general equation

• \((x - h)^2 + (y - k)^2 = r^2\)

The \((h, k)\) represents the center of the circle. If you think of a circle as being like a perfectly round pizza, the center is where you would place the hole for spinning it around.
In our example, the circle's center is at the origin, which means \(h = 0\) and \(k = 0\). This places the circle right in the center of the coordinate plane, which is a special and common location for the center of a circle in many problems to simplify calculations.
When the center is at the origin, the equation simplifies significantly, eliminating the \(h\) and \(k\) terms, thus reducing our equation to:

• \(x^2 + y^2 = r^2\)

Understanding the center is integral because any change in its location affects the entire placement of the circle in the graph.

Radius of a Circle

The radius of a circle plays a fundamental role in determining the size and scale of the circle. In the general circle equation

• \((x - h)^2 + (y - k)^2 = r^2\)

The radius \(r\) is represented as the distance from the center \((h, k)\) to any point \((x, y)\) on the circle boundary.
In simple terms, imagine you draw a line from the center of the circle to any point on its edge—that line is your radius. In our specific example, the radius is 2, meaning the circle measures 2 units from the center in all directions.
To use this in the general equation, you substitute \(r\) with its value, squaring it for the equation. This results in the specific form:

• \(x^2 + y^2 = 4\)

The value of the radius not only affects the size but also shows how the function represents the circle's extent on the plane. The larger the radius, the larger the circle.