Problem 12 - A Graphical Approach to College Algebra

Question

Find the center-radius form for each circle satisfying the given conditions. Center \((0,0) ;\) radius 5

Step-by-Step Solution

Answer

The equation of the circle is \(x^2 + y^2 = 25\).

Step 1: Recall the Center-Radius Form of a Circle

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

Step 2: Identify the Values for Center and Radius

From the problem, the center of the circle is given as \((0, 0)\), and the radius is given as \(5\).

Step 3: Substitute the Center and Radius into the Equation

Substitute \(h = 0\), \(k = 0\), and \(r = 5\) into the center-radius form equation. This gives \((x-0)^2 + (y-0)^2 = 5^2\).

Step 4: Simplify the Equation

Simplify the equation by eliminating zeroes and calculating the square of the radius: \(x^2 + y^2 = 25\).

Key Concepts

Center-Radius Form

Understanding the center-radius form of a circle is essential when working with circle equations. This form is a special equation that provides the relationship between the coordinates of any point on the circle and the fixed center point. It is expressed as:\[(x-h)^2 + (y-k)^2 = r^2\]In this equation:

• \((h, k)\) represents the circle’s center coordinates.
• \(r\) is the circle's radius.
• The variables \(x\) and \(y\) are the coordinates of a point located on the circle itself.

The beauty of this equation is in its simplicity and directness. By simply substituting the center's coordinates and radius into the formula, you can describe any circle perfectly. For example, a circle centered at the origin, \((0,0)\), with a radius of \(5\), uses this form to precisely convey its geometry.

Circle Geometry

Circle geometry is a captivating topic in mathematics, focusing on the properties and measurements of circles. At the heart of circle geometry is understanding how all the points are equidistant from a central point.Key aspects of circle geometry include:

• **Radius:** This is the distance from the center of the circle to any point on its boundary. All radii in a circle are equal.
• **Diameter:** Equal to twice the radius, the diameter is the longest distance across the circle.
• **Circumference:** This is the total distance around the circle, calculated using the formula \(C=2\pi r\), where \(r\) is the radius.
• **Area:** Representing the space contained within a circle, this is calculated as \(A=\pi r^2\).

Understanding these aspects helps in visualizing and solving problems involving circles, such as determining where a point lies relative to a given circle or calculating distances.

Algebraic Equations

Algebraic equations form the backbone of expressing mathematical relationships, including those in circle geometry. In the case of the center-radius form of a circle, the equation is algebraic because it consists of variables that capture the relationship between the circle's center, its radius, and any point on the circle.This form is algebraic because:

• It involves two variables, \(x\) and \(y\), representing coordinates in a plane.
• By squaring the differences \((x-h)\) and \((y-k)\), it ensures all points equidistant from the center \((h, k)\) fall on the circle perimeter.
• The equation is quadratic, a type of polynomial equation, which frequently appears in algebra.

Equations like these are essential for solving complex problems. They help describe the circle in a plane and are crucial when translating geometrical figures into understandable numeric forms. This allows us to analyze and predict circle behavior in both mathematical and real-world scenarios.