Problem 3 - Algebra 2

Question

Write an equation of a circle with the given center and radius. Check your answers. $$ (2,3), 4.5 $$

Step-by-Step Solution

Answer

The equation of the circle with center coordinate (2,3) and radius 4.5 is \( (x-2)^2 + (y-3)^2 = 20.25 \).

Step 1: Understanding the Circle Equation

Firstly, the general formula for the equation of a circle can be determined with the formula \( (x-h)^2 + (y-k)^2 = r^2 \), where (h,k) denotes the center of the circle and r represents the radius.

Step 2: Substitute the values into the circle equation

Substitute the center coordinate (h,k) and radius r into the circle equation. In this case, h=2, k=3 and r=4.5. Therefore, the equation becomes \( (x-2)^2 + (y-3)^2 = (4.5)^2\) .

Step 3: Simplification of the Equation

Simplify the right-hand side of the equation where \( (4.5)^2=20.25\). Therefore, the equation results as \( (x-2)^2 + (y-3)^2 = 20.25 \).

Key Concepts

Circle Equation

The equation of a circle is an algebraic expression used to describe all the points that form a perfect circle on a coordinate plane. Circles are defined by a set of points that are equidistant from a fixed center point. This is captured mathematically by the equation \( (x-h)^2 + (y-k)^2 = r^2 \). Here, \((h, k)\) represents the center of the circle, and \(r\) denotes the radius.
This standard form equation is derived based on the definition of a circle, making it flexible enough to apply to any circle drawn on the xy-plane. Each part of the equation has its significance in identifying the complete form and structure of any circle.

Center of a Circle

The center of a circle is a crucial part of its equation as it determines the circle's precise location on a plane. In the circle equation \((x-h)^2 + (y-k)^2 = r^2\), the coordinates \((h, k)\) indicate the center.
Understanding the center:

• The variable \(h\) represents the horizontal shift from the origin, whereas, \(k\) represents the vertical shift.
• This means if \(h = 2\) and \(k = 3\), the center of the circle is at point \((2, 3)\).

The knowledge of where the center lies helps in drawing or visualizing the circle accurately on the coordinate axes. Visualizing the center allows students to better understand how different transformations affect the position of the circle.

Radius of a Circle

The radius of a circle is the distance from its center to any point on its circumference. For the circle equation \((x-h)^2 + (y-k)^2 = r^2\), the \(r\) in the equation stands for this radius.
Importance of the radius includes:

• It is a non-negative number that affects the size of the circle. A larger radius results in a larger circle.
• For example, if \(r = 4.5\), the radius of the circle is 4.5 units.

Squaring the radius to form \(r^2\) is vital in the equation because it ensures that every term remains non-negative, allowing for accurate representation of all circle sizes on the graph.

Algebraic Expressions

Algebraic expressions form the backbone of forming and solving equations, such as those found with circles. They consist of numbers, variables, and operations such as addition, subtraction, multiplication, and division.
In the context of circle equations:

• Expressions like \((x-h)^2\) and \((y-k)^2\) are algebraic terms used to denote the squared differences between any point \((x, y)\) and the circle’s center \((h, k)\).
• Simplifying expressions is often necessary, as seen when transforming the equation from \((x-2)^2 + (y-3)^2 = (4.5)^2\) to \((x-2)^2 + (y-3)^2 = 20.25\), where \((4.5)^2 = 20.25\).

This simplification helps to achieve an easier-to-read and manipulate form, which is particularly important when solving complex algebraic problems or graphing the circle accurately.