Problem 71 - Algebra 2

Question

Write an equation of a circle with the given center and radius. center \((0,0),\) radius 8

Step-by-Step Solution

Answer

The equation of the circle is \(x^2 + y^2 = 64\).

Step 1: Identify the given information

We are given that the center of the circle is at the origin \((0,0)\) and the radius of the circle is 8. This means that the radius squared, \(r^2\), is \(8^2 = 64\).

Step 2: Substitute the radius into the equation of the circle

Substitute the radius which we've found was 64 into the general equation of the circle. This brings us to \(x^2 + y^2 = 64\).

Step 3: Write the final equation of the circle

From substituting into the general equation, we obtain the desired equation of our circle as \(x^2 + y^2 = 64\).

Key Concepts

Geometry

In geometry, a circle is a set of points that are all at the same distance from a central point. This distance is known as the radius. A circle is a two-dimensional shape, meaning it has length and width but no depth. To fully understand how a circle works, it's crucial to examine its defining features and how they relate to each other.

• A circle can be described using an equation, especially in a coordinate plane.
• The equation helps us understand concepts like the circle's center, radius, and the relationship between its points.

Understanding the basic geometry of a circle lays the groundwork for solving problems involving circle equations.

Coordinate System

A coordinate system is a method for identifying the position of points in space. In the plane of a circle, we use the Cartesian coordinate system, where each point has an "x" and a "y" coordinate. These coordinates help in visualizing and solving geometric problems.

• The center of the circle is represented as a point \( (h, k) \) in this system.
• The standard equation for a circle in a coordinate plane is \( (x - h)^2 + (y - k)^2 = r^2 \).
• This equation tells us that any point \( (x, y) \) on the circle's edge has the same distance r from the center.

The coordinate system is a powerful tool for dealing with circular shapes mathematically.

Radius

The radius of a circle is the distance from its center to any point on its boundary. In our exercise, the radius is specifically given as 8.

• The radius squared, represented as \( r^2 \), is found by multiplying the radius by itself (\( 8^2 = 64 \)).
• The radius is crucial since it directly affects the size of the circle and is a key part of the circle's equation.

A clear understanding of the radius makes it easier to write and manipulate the equation of a circle.

Center of a Circle

The center of a circle is the point from which every point on the circle is equidistant. For our circle, the center is at the origin, \( (0, 0) \).

• In the circle's equation \( x^2 + y^2 = 64 \), the center \( (h, k) \) is at \( (0, 0) \).
• When the center is at the origin, the equation simplifies because the terms involving h and k vanish, making calculations simpler.

The center is a vital part of understanding the equation of the circle and helps determine how the circle is positioned in the plane.