Problem 9 - A Graphical Approach to College Algebra

Question

Find the center-radius form for each circle satisfying the given conditions. Center \((1,4) ;\) radius 3

Step-by-Step Solution

Answer

The center-radius form of the circle is \((x - 1)^2 + (y - 4)^2 = 9\).

Step 1: Understand the Center-Radius Form Equation

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

Step 2: Identify the Given Values

From the problem, we know that the center of the circle is \((h, k) = (1,4)\) and the radius \(r = 3\).

Step 3: Substitute the Values into the Equation

Substitute the values of \(h\), \(k\), and \(r\) into the center-radius equation, where \(h = 1\), \(k = 4\), and \(r = 3\).

Step 4: Write the Center-Radius Form

Plugging these values into the equation \((x - h)^2 + (y - k)^2 = r^2\) gives us \((x - 1)^2 + (y - 4)^2 = 3^2\).

Step 5: Simplify the Equation

Calculate \(3^2\) to get 9, so the equation becomes \((x - 1)^2 + (y - 4)^2 = 9\).

Key Concepts

Equation of a Circle

The equation of a circle in the center-radius form is a straightforward algebraic expression.
It allows us to easily describe all the points (that satisfy this equation) that make up the circle.
The general form of the center-radius equation is given by: \[(x - h)^2 + (y - k)^2 = r^2\]Here:

• \((h, k)\) represents the center of the circle.
• \(r\) is the radius of the circle.

The equation states that the distance from any point on the circumference to the center is always equal to the radius. By substituting the values of \((h, k)\) and \(r\) into this formula, you can find the specific equation for a particular circle.

Circle Geometry

Circle geometry revolves around understanding various properties and characteristics of circles.
A fundamental part of this geometry includes identifying aspects such as the center, radius, and diameter.
Here are some essential points:

• The center is the middle point of the circle, around which all points on the circle are symmetrically placed.
• The radius is the distance from the center to any point on the circle. It is consistent throughout.
• The diameter is twice the radius and passes through the center, touching two points on the circle.

These fundamental notions help in understanding and solving various problems related to circles, such as calculating the area and circumference as well.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols.
In the context of the equation of a circle, understanding algebraic expressions allows for manipulation and solving of the circle equation.
Key elements include:

• Variables, like \(x\) and \(y\), which denote the coordinates of any point on the circle.
• Operations (addition, subtraction, and squares) that define the relationship between the center and any point on the circumference.
• Equating the expression to the square of the radius, which sets the circle's boundaries.

By mastering algebraic expressions, we gain the skills needed to deconstruct and understand circle-related equations fully.