Problem 34 - College Algebra

Question

Write the standard form of the equation of the circle with the given center and radius. $$\text { Center }(2,-1), r=4$$

Step-by-Step Solution

Answer

The standard form of the equation of the circle is \((x-2)^2 + (y+1)^2 = 16\).

Step 1: Recognize the standard form of a circle

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: Substitute given values into the standard form

The center (2, -1) and radius 4 can be substituted into the equation. This gives us \((x-2)^2 + (y+1)^2 = 4^2\).

Step 3: Simplify the equation

The equation can be simplified by calculating the square of 4 which equals 16. So the standard form of the equation becomes \((x-2)^2 + (y+1)^2 = 16\).

Key Concepts

Standard Form

The standard form of the equation of a circle is a straightforward, yet powerful formula that allows you to precisely describe any circle on a Cartesian plane. The general formula looks like this:

• \((x-h)^2 + (y-k)^2 = r^2\)

In this formula:

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

Using this structure, you're able to plug in specific values that represent the circle you are working with. This helps in identifying the size and location of the circle in a clear, mathematical way. The standard form is particularly useful because it aligns neatly with the circle's geometric properties, providing a direct link between algebra and geometry.
This formula simplifies many problems related to circles, as it captures all the essential information needed in a compact equation.

Center and Radius

Understanding the center and radius of a circle is crucial when working with its equation. These two pieces of information essentially define the circle:

• **Center:** The center \((h, k)\) is the point around which the circle is perfectly symmetrical. In an equation, these are the values you substitute for \(h\) and \(k\).
• **Radius:** The radius \(r\) is the distance from the center to any point on the circle's edge. This distance dictates how large the circle is.

Given the center \((2, -1)\) and radius \(4\) in this exercise, these values plug directly into the standard form formula. So your equation begins as \((x-2)^2 + (y+1)^2 = 4^2\), showing how the center and radius are the foundational elements of a circle's equation. They work together to map out the precise shape and location of the circle you are trying to describe.

Circle Equation

A circle's equation in standard form not only helps describe the circle but also offers an easy way to perform calculations or transformations. After inserting the center and radius into the standard form, as shown in this example, the equation is almost complete:

• Start by plugging in the center: \((x-2)^2 + (y+1)^2\)
• Then include the radius: \(= 4^2\)
• Simplify further by squaring the radius: \((x-2)^2 + (y+1)^2 = 16\)

This equation tells us everything we need to know about the circle. By using a squared radius, the equation emphasizes that all circle points are equally distant from the center. Additionally, having both sides of the equation equal makes it easy to verify points that lie directly on the circle. This type of equation makes mathematical tasks, like finding intersections or transforming the circle, much more manageable.