Problem 40 - College Algebra Essentials

Question

write the standard form of the equation of the circle with the given center and radius. $$ \text { Center }(-2,0), r=6 $$

Step-by-Step Solution

Answer

The standard form of the equation of the circle is \((x+2)^2 + y^2 = 36\).

Step 1: Identify the Given Information

The center of the circle, represented by the ordered pair (-2,0), means that h = -2 and k = 0. The radius of the circle is given as r = 6.

Step 2: Substitute Given Values into Circle Equation

Substitute the given values h = -2, k = 0, and r = 6 into the standard form of the equation of a circle \((x-h)^2 + (y-k)^2 = r^2\). The resulting equation is therefore \((x-(-2))^2 + (y-0)^2 = (6)^2\).

Step 3: Simplify the Equation

Simplify the equation obtained in Step 2. The equation simplifies to \((x+2)^2 + y^2 = 36\).

Key Concepts

Standard Form

The standard form of a circle equation is an extremely useful algebraic expression. It allows us to understand the geometric properties of a circle easily. This form is given as \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle, and \(r\) is its radius. The beauty of using the standard form lies in its simplicity and clarity. By just looking at the equation, one can immediately identify key features of the circle: its position and size. Breaking down the components:

• \(x\) and \(y\) are the variables that represent coordinates on a plane.
• \(h\) and \(k\) are constants that indicate the center's position.
• \(r\) is a constant that represents the circle's radius.

Understanding this format is fundamental to working with circle equations.
Using substitution to plug values into the standard form can help solve many geometric problems clearly and concisely.

Circle Center

The center of a circle is a crucial detail when defining or working with circles in algebra. It's the fixed point from which every point on the circumference is equidistant.
In the standard equation \((x-h)^2 + (y-k)^2 = r^2\), the coordinates \((h, k)\) signify the circle's center:

• \(h\) represents the x-coordinate of the center.
• \(k\) represents the y-coordinate of the center.

The center affects where the circle is positioned on the coordinate plane. For example, if the center is at \((-2, 0)\), this means the circle is shifted 2 units to the left along the x-axis from the origin. Identifying and understanding the circle center allows one to easily graph or visualize a circle.

Circle Radius

The radius of the circle is another key parameter that determines the size of the circle. In the standard circle equation \((x-h)^2 + (y-k)^2 = r^2\), the value \(r\) represents the radius.
The radius is the distance from the center of the circle to any point on its edge. It's a crucial element for defining the scale of the circle:

• The square of the radius is the constant on the right side of the equation \(r^2\).
• Therefore, if \(r = 6\), then substituting into the equation gives us \(r^2 = 36\).

This value informs us how large or small the circle is, making it easy to draw or interpret accurately on a graph.

Algebra

Algebra plays an essential role in working with circle equations. It lets us manipulate and rearrange expressions to simplify or solve problems. When given the center and radius, algebra helps in forming the standard equation of a circle.
Given our exercise, we use algebraic substitution:

• Start with the standard form \((x-h)^2 + (y-k)^2 = r^2\).
• Substitute the center \((-2, 0)\) and radius \(6\) into the equation.
• Simplifying \((x-(-2))^2 + (y-0)^2 = 6^2\) transforms it to \((x+2)^2 + y^2 = 36\).

This process of substitution and simplification helps maintain accuracy and clarity.Handling circle equations through algebra is foundational. It provides a clear method for expressing geometrical concepts algebraically, making complex information about circles accessible.