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Chapter 10: Problem 138

Find the equation of a circle satisfying the given conditions. Center: (0,0)\(;\) radius: 9

Short Answer

Expert verified
x^2 + y^2 = 81

Step by step solution

01

General Equation of a Circle

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

Substitute the Center

Substitute the given center \( (0, 0) \) into the general formula. Since h = 0 and k = 0, the equation simplifies to: \[ (x - 0)^2 + (y - 0)^2 = r^2 \]
03

Substitute the Radius

Substitute the given radius \( r = 9 \) into the equation. This changes the equation to: \[ x^2 + y^2 = 9^2 \]
04

Simplify the Equation

Simplify the equation by squaring the radius: \[ x^2 + y^2 = 81 \]

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Key Concepts

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

center of a circle
The center of a circle is a specific point that serves as the middle of the circle. It is often denoted by the coordinates \( (h, k) \). Every point on the circle is equidistant from this central point. When you know the center and the radius, you can easily define the circle's equation.

In the example provided, the center is given as \( (0, 0) \). This means the circle is centered at the origin of the coordinate system.

To find the equation of the circle, you substitute the center鈥檚 coordinates into the general formula for the equation of a circle. This simplifies the math and helps focus on understanding the relationship between the circle's center and its radius.
radius of a circle
The radius of a circle is the distance from the center to any point on the circle itself. It is a constant value for a given circle and is typically denoted by the letter \ r \. Knowing the radius is essential for finding the equation of the circle.

In our example, the radius is provided as 9.

Substituting 9 into the formula makes it possible to simplify the equation. By squaring the radius ( \( r^2 \) ), you get the term that completes the circle's equation: \( x^2 + y^2 = 81 \). \

This showcases the straightforward relationship between the radius and the circle's overall equation, solidifying why it is a crucial component in working out problems involving circles.
general form of a circle
The general form of the equation of a circle provides a unified way to express any circle mathematically. The formula is given by: \( (x - h)^2 + (y - k)^2 = r^2 \).

Here:
  • \( (h, k) \) represents the coordinates of the center of the circle.
  • \ r \ is the radius of the circle.
In our example, substituting the center \( (0, 0) \) simplifies the formula to \( (x - 0)^2 + (y - 0)^2 = r^2 \), which further simplifies to \( x^2 + y^2 = r^2 \).

Next, substituting the radius \ r = 9 \ into the equation shows its simplification to \( x^2 + y^2 = 81 \).

This demonstrates how powerful and flexible the general equation is, making it a fundamental tool in geometry for defining and analyzing circles.

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Most popular questions from this chapter

Simplify \(-\sqrt[4]{1250}\)

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List all of the following sets to which each number belongs. A number may belong to more than one set. real numbers pure imaginary numbers nonreal complex numbers complex numbers $$ \sqrt{-49} $$
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