Problem 11 - A Graphical Approach to College Algebra

Question

Find the center-radius form for each circle satisfying the given conditions. Center \((0,0) ;\) radius 1

Step-by-Step Solution

Answer

The circle equation is \(x^2 + y^2 = 1\).

Step 1: Understand Circle Equations

The general equation for a circle in center-radius form 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 the Center Coordinates

We are given the center \((0,0)\), so substitute \(h=0\) and \(k=0\) into the equation: \((x-0)^2 + (y-0)^2 = r^2\). Simplifying, we get \(x^2 + y^2 = r^2\).

Step 3: Substitute the Radius

The radius is given as 1. Substitute \(r=1\) into the simplified equation from Step 2: \(x^2 + y^2 = 1^2\).

Step 4: Simplify the Equation

Simplify the equation \(x^2 + y^2 = 1^2\) to get the final form: \(x^2 + y^2 = 1\). This is the center-radius form for the circle with the given conditions.

Key Concepts

center-radius form

The center-radius form is a specific way to express the equation of a circle. It's a very handy format because it highlights the circle's most essential properties: its center and its radius.
A circle in the center-radius form is written as

• \[(x-h)^2 + (y-k)^2 = r^2\]

Here, - \(h\) and \(k\) are the coordinates of the circle's center, - and \(r\) represents the radius.
This form is invaluable in geometry because it makes identifying the circle's center and radius straightforward.

radius of a circle

The radius of a circle is one of its defining characteristics. Imagine a line stretching from the center of the circle to any point on the circumference. That's the radius!
The circle's radius is crucial because it determines the circle's size. It's the distance that all points on the edge of the circle maintain from the center.
When we have a circle equation in the center-radius form

• like \(x^2 + y^2 = 1\),

we can clearly see that the radius is the square root of the number on the right side of the equation.
For example, in \(x^2 + y^2 = 1\), the radius is \sqrt{1}\, which is simply 1.

coordinates of a circle center

The coordinates of a circle's center are simply a pair of numbers that indicate where the center of the circle is placed on the coordinate plane. These coordinates are an essential part of the circle's equation in center-radius form.
In expressions like

• \[(x-h)^2 + (y-k)^2 = r^2\],

The \(h\) and \(k\) are the coordinates of the center, represented as \((h, k)\).
They tell you exactly where the circle is located. For example, a circle with a center at \(0,0\) is located right at the origin of the coordinate plane.
Understanding these coordinates helps position the circle precisely on any graph and allows further geometric or algebraic operations.