Problem 33 - Intermediate Algebra

Question

Write the equation of a circle in standard form with the following properties. Center at the origin; diameter \(4 \sqrt{2}\)

Step-by-Step Solution

Answer

The standard form equation is \(x^2 + y^2 = 8\).

Step 1: Understanding the Circle Equation

The standard form of a circle's equation is \[(x - h)^2 + (y - k)^2 = r^2\]where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Identify the Center

The problem states that the circle's center is at the origin, thus \((h, k) = (0, 0)\). Substituting these into the equation, our equation becomes \[x^2 + y^2 = r^2.\]

Step 3: Calculate the Radius

The diameter of the circle is given as \(4 \sqrt{2}\). The radius is half of the diameter, so we divide by 2.\[r = \frac{4 \sqrt{2}}{2} = 2 \sqrt{2}.\]

Step 4: Substitute the Radius

Now substitute the radius we found into the circle equation:\[x^2 + y^2 = (2 \sqrt{2})^2.\] Simplifying,\[x^2 + y^2 = 8.\]

Key Concepts

Standard Form of a Circle

The equation of a circle in standard form is a key concept in geometry, one that is easy to grasp with a clear understanding of its structure. The standard form is given by the equation

• \((x - h)^2 + (y - k)^2 = r^2\),

where \((h, k)\) represents the coordinates of the circle's center, and \(r\) is the radius of the circle.

When writing a circle equation in standard form, it's essential to identify where all the components fit. The variables \(x\) and \(y\) allow the equation to adapt to any location in a Cartesian coordinate system by shifting it according to \(h\) and \(k\) without changing its size.

In the exercise, since the center is at the origin \((0, 0)\), the equation simplifies beautifully to

• \(x^2 + y^2 = r^2\).

This is one of the simplest forms of a circle equation, showing symmetry about the origin.

Circle Center

The center of a circle is immensely important as it defines the circle's position in the coordinate plane. It's denoted by the point \((h, k)\) in the standard circle equation.

• At \((0, 0)\), the circle is rooted at the origin, aligning perfectly along the x and y axes.
• If \(h\) or \(k\) are nonzero values, the circle shifts along the axes to accommodate those values.

Determining the center, as was provided in this exercise, as \((0, 0)\), is crucial because it directly influences how the circle's equation is structured.This simplification to the origin results in

• \(x^2 + y^2 = r^2\).

Hence, every alteration in the coordinates \((h, k)\) results in repositioning the circle without altering its size, only its location.

Radius Calculation

Calculating the radius, especially from a given diameter, is crucial in forming the circle's equation. The radius \(r\) is essentially half the diameter, as a circle's diameter is twice its radius. This relationship is expressed as:

• \(r = \frac{d}{2}\),

where \(d\) stands for the diameter.
Once the radius is determined, it squares in the equation

• \((x - h)^2 + (y - k)^2 = r^2\)

to define the circle's size.

In this exercise, with a diameter of \(4 \sqrt{2}\), the radius calculates as

• \(2 \sqrt{2}\).

Square this value for the equation, resulting in the final equation:

• \(x^2 + y^2 = 8\),

leading to a clear visualization of the circle.