Problem 26 - Intermediate Algebra

Question

Write the equation of a circle in standard form with the following properties. Center at \((5,3) ;\) radius 2

Step-by-Step Solution

Answer

The equation of the circle is \\( (x - 5)^2 + (y - 3)^2 = 4 \\).

Step 1: Identify the Standard Form Formula for a Circle

The standard form of the equation of a circle with center \(h, k\) and radius \ r \ is given by \( (x - h)^2 + (y - k)^2 = r^2 \). Here, \( (h, k) = (5, 3) \), and \( r = 2 \).

Step 2: Substitute the Center Coordinates

Plug the center coordinates \( h = 5 \) and \( k = 3 \) into the circle equation to get \( (x - 5)^2 + (y - 3)^2 = r^2 \).

Step 3: Substitute the Radius

Replace \( r \) with the radius value, which is 2. This modifies the equation to \( (x - 5)^2 + (y - 3)^2 = 2^2 \).

Step 4: Simplify the Equation

Calculate \( 2^2 \) which equals 4. Finalizing the equation gives \( (x - 5)^2 + (y - 3)^2 = 4 \). This is the standard form of the circle's equation.

Key Concepts

Standard Form

In the world of mathematics, particularly in geometry, the standard form of a circle's equation is a very useful tool. This form allows us to clearly see both the center and radius of a circle from its equation. The standard form is given by:

• \((x - h)^2 + (y - k)^2 = r^2\)

In this formula, \(h\) and \(k\) represent the x and y coordinates of the circle's center, while \(r\) is the circle's radius. When a circle's equation is written in this format, it is easy to identify these properties directly from the equation. For example, using the standard form is helpful because it immediately lets us know where the circle is positioned on a coordinate plane and how large the circle is. This format forms the backbone for graphing circles accurately and efficiently in coordinate geometry.

Center and Radius

Every circle has specific characteristics, namely its center and radius, which determine its size and position on a plane.

• The **center** of the circle, denoted as \( (h, k) \), indicates the exact middle point of the circle. This point is equidistant from any point on the circle's boundary.
• The **radius** \( r \) is the distance from the center to any point on the circumference of the circle. It essentially determines the size of the circle.

Once these values are known, they can be directly inserted into the standard form equation for circles: \( (x - h)^2 + (y - k)^2 = r^2 \). As illustrated in the original exercise, if a circle has a center at \( (5, 3) \) and a radius of 2, substituting these values into the equation gives:

• Center: \( h = 5 \, k = 3 \)
• Radius: \( r = 2 \)

Therefore, the circle's equation in standard form becomes: \( (x - 5)^2 + (y - 3)^2 = 2^2 \), which simplifies to \( (x - 5)^2 + (y - 3)^2 = 4 \). This equation immediately reveals the circle's core properties, making it easy to graph or analyze further.

Geometry

Geometry helps us understand the spatial arrangement and properties of shapes, and circles are a fundamental shape studied in this field. Circles have unique symmetry and uniformity, and their equations are deeply rooted in geometric principles. By studying the equation of a circle in standard form, we can easily determine key attributes like position, size, and relationships within a coordinate plane.
Using the standard form, one can explore:

• The concept of **symmetry** around the center point, where every point on the circle is at an equal distance from its center.
• How varying the radius \( r \) affects the circle's size and space it occupies on the graph.
• The impact of shifting the center \( (h,k) \), which moves the entire circle across the plane without altering its shape.

Understanding these geometric principles allows students to predict how changes in the circle's equation will affect its graph and properties. This foundational concept enables further exploration into more complex geometric constructions and transformations.