Problem 32 - College Algebra Essentials

Question

write the standard form of the equation of the circle with the given center and radius. $$ \text { Center }(0,0), r=8 $$

Step-by-Step Solution

Answer

The standard form of the equation of the circle with center at the origin (0,0) and radius 8 is \( x^2 + y^2 = 64 \).

Step 1: Identify the parameters of the circle

Identify the parameters of the circle, which are given as follows: center at the origin which is (0,0), thus a = 0 and b = 0, and radius r = 8.

Step 2: Substitute the parameters into the circle equation

Substitute the parameters into the standard form of the equation of a circle, which is \((x-a)^2 + (y-b)^2 = r^2\). Substituting a = 0, b = 0, and r = 8 gives \( (x-0)^2 + (y-0)^2 = 8^2 \).

Step 3: Simplify the equation

Simplify the equation to its simplest form. The simplified form of the equation is \( x^2 + y^2 = 64 \)

Key Concepts

Circle Parameters

Understanding the parameters of a circle is crucial when dealing with its equations. A circle can be completely defined by two key elements: its center and its radius. The 'center' is a point represented by coordinates (a, b), which indicates the circle's position on the coordinate plane. On the other hand, the 'radius' is a single number, r, that describes the size of the circle, which is the constant distance from the center to any point on the circle.

When given the center (0,0) and radius 8, as in the exercise, these values are essential in formulating the circle's equation. A good tip is to always double-check these parameters to avoid any possible errors when forming the equation.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, bridges algebra and geometry, using a coordinate system to visually represent algebraic equations. Points are located using ordered pairs, such as (x, y). In the context of circles, coordinate geometry is vital as it allows us to visualize the position of the circle on the plane and understand the circle's relationship to other geometric shapes.

In the standard form equation of a circle, (x-a)^2 + (y-b)^2 = r^2, a and b represent the x and y coordinates of the center, respectively. Thus, the equation is a portrayal of all the points (x, y) that are at a radius' distance from the center, creating a perfect circle on the plane. Remember to visualize the problem on graph paper if you're struggling to understand the concept, as seeing the circle can make it easier to grasp.

Algebraic Equations

Algebraic equations are the foundation for expressing mathematical relationships symbolically. In the context of a circle's equation, we are dealing with a quadratic equation in both x and y because of the squared terms. The beauty of algebra is in how it simplifies complex relationships into manageable expressions.

For instance, the exercise’s circle with a center at (0,0) and radius 8 can be expressed as x^2 + y^2 = 64. This is the algebraic way to represent all the points that are exactly 8 units away from the origin—which we know forms a circle. Grasping how to manipulate these equations is key to successfully solving geometry problems involving circles. It's important to practice simplifying these equations by squaring the radius and setting the equation to zero to get a fuller understanding of their implications in geometry.