Problem 19 - Beginning and Intermediate Algebra

Question

Find an equation of the circle with the given center and radius. Center \((-3,2) ;\) radius \(=1\)

Step-by-Step Solution

Answer

The equation of the circle with the given center \((-3, 2)\) and radius \(1\) is \((x + 3)^2 + (y - 2)^2 = 1\).

Step 1: Write down the general equation of a circle.

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

Step 2: Substitute the center coordinates.

The given center coordinates are \((-3, 2)\). So, we can replace \(h\) with \(-3\) and \(k\) with \(2\). The equation becomes: \((x - (-3))^2 + (y - 2)^2 = r^2\).

Step 3: Substitute the radius value.

The given radius is \(1\). We can replace \(r\) with \(1\) in the equation. So, the equation becomes: \((x - (-3))^2 + (y - 2)^2 = 1^2\).

Step 4: Simplify the equation.

We can simplify the equation by removing the parentheses and squaring the radius: \((x + 3)^2 + (y - 2)^2 = 1\). The equation of the circle with the center at \((-3, 2)\) and radius equals to 1 is: \((x + 3)^2 + (y - 2)^2 = 1\).

Key Concepts

Circle Equation

The equation of a circle is a simple algebraic formula that describes all the points making up the circle on a coordinate plane. It's expressed in the format \[(x - h)^2 + (y - k)^2 = r^2\]Here,

• \( (h, k) \) represent the center of the circle. They are the coordinates for the exact middle point of the circle.
• \( r \) stands for the radius. It is the distance from the center to any point on the circle.

Each solution to this equation corresponds to a unique point lying on the circle's circumference. Substituting the values of \( h \), \( k \), and \( r \) in the equation gives the specific equation for a particular circle. This concise formula allows us to explore various properties of circles easily.
It makes solving geometry problems related to circles much more straightforward by providing a clear and easy-to-use framework.

Radius and Center of a Circle

To define a circle on a graph, you need the center point and the radius. The center is a fixed point in the plane, and the radius is the length measured from this point to the edge of the circle.

• **Center**: The center of a circle is typically represented by coordinates \( (h, k) \). For a circle with a center of \((-3, 2)\), the circle is centered at 3 units left and 2 units up on a Cartesian plane. This is the heart of the circle from which the circle forms.
• **Radius**: The radius is a line segment between the center of the circle and any point on its edge. A radius of \(1\) means every point on the circle is exactly 1 unit away from its center.

Knowing the center and radius is crucial as each value determines the circle's size and position on a coordinate plane. These two elements give the circle its definition and allow us to write its equation using the circle's equation formula.

Algebraic Equations

Algebraic equations are calculations involving variables and constants. They are the heart of mathematical problem-solving and help us express relationships between different quantities.
Let's consider how the circle equation is an example of an algebraic equation: it uses constants (like the radius) and variables (like \(x\) and \(y\), representing unknown points on the circle).

• **Variables**: Represents unknowns that we aim to solve or describe. In the circle's equation, \(x\) and \(y\) depict any point's possible coordinates on the circle's edge.
• **Constants**: These are fixed values; in our circle equation, \(h\), \(k\), and \(r\) serve as constants. They give specific characteristics to the circle, such as its position (center) and size (radius).

Understanding algebraic equations allows students to solve for unknown variables and understand the geometric relationships they describe on a plane. They are foundational in many areas of math, making them indispensable in both academic learning and real-world applications.