/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 91 Find the equation of a circle wi... [FREE SOLUTION] | 91影视

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Chapter 0: Problem 91

Find the equation of a circle with center (0,0) and radius 1 .

Short Answer

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

Step by step solution

01

Identify the General Form of the Circle Equation

The general form of the equation of a circle with center \(h, k\) and radius \(r\) is given by the formula \[ (x - h)^2 + (y - k)^2 = r^2 \]
02

Substitute the Center Coordinates

In this problem, the center of the circle is given as \(0,0\). So, substitute \(h = 0\) and \(k = 0\) into the general formula. This gives \[ (x - 0)^2 + (y - 0)^2 = r^2 \]
03

Simplify the Equation with the Radius

The radius \(r\) is given as 1. Substitute \(r = 1\) into the simplified equation. This results in \[ x^2 + y^2 = 1 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

circle equation
The equation of a circle is one of the core concepts in geometry, depicting the set of all points that are at a fixed distance from a given point, known as the center. The general form of the equation of a circle with center at \(h, k\) and radius \(r\) is given by the formula \[ (x - h)^2 + (y - k)^2 = r^2 \] In this formula, \(x\) and \(y\) represent the coordinates of any point on the circle, while \(h\) and \(k\) are the coordinates of the circle's center. The term \(r^2\) represents the square of the radius. By applying this formula, one can easily find the equation of a circle when the center and radius are known.
geometry
Geometry is a branch of mathematics that explores shapes, sizes, and the properties of space. Within geometry, the concept of a circle is fundamental. A circle is defined as the set of all points in a plane that are a fixed distance (the radius) from a given point (the center). This definition leads us to the equation of the circle. Understanding the elements that make up a circle鈥攕uch as the center and radius鈥攊s crucial in solving problems related to circles. In our exercise, the center is at (0,0), and the radius is 1, thus we use the circle equation accordingly.
radius
The radius of a circle plays a crucial role in its equation. The radius is the fixed distance from the center of the circle to any point on the circle. In the equation \[ (x - h)^2 + (y - k)^2 = r^2 \], the term \(r^2\) represents the square of the radius. For example, if the radius is given as 1, we substitute \(r = 1\) into the formula, resulting in \[ x^2 + y^2 = 1 \] This equation tells us that for any point \(x, y\) on the circle with center (0,0) and radius 1, the sum of the squares of the coordinates is equal to 1. The radius is a key measure that impacts the size and scale of the circle, influencing its overall shape and position in the coordinate plane.

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