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Chapter 5: Problem 4

Write an equation of the circle centered at (-9,9) with radius 16 .

Short Answer

Expert verified
The equation of the circle is \((x + 9)^2 + (y - 9)^2 = 256\).

Step by step solution

01

Identify the Standard Equation of a Circle

The standard equation of a circle in the Cartesian coordinate system is given by \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is its radius.
02

Substitute the Center Coordinates

Identify the circle's center coordinates \((-9, 9)\), then substitute them into the formula. This gives \((x + 9)^2 + (y - 9)^2 = r^2\).
03

Substitute the Radius

The radius of the circle is 16. Substitute \(r = 16\) into the equation. This results in \((x + 9)^2 + (y - 9)^2 = 16^2\).
04

Simplify the Equation

Calculate \(16^2\) to simplify the equation: \(16^2 = 256\). This gives the final equation \((x + 9)^2 + (y - 9)^2 = 256\).

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Key Concepts

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

Cartesian Coordinate System
The Cartesian coordinate system is a method of representing geometric figures through a series of numerical coordinates along two perpendicular axes.
These axes are typically labeled as the x-axis and the y-axis. Each point in this system is identified by a pair of numbers, \(x, y\).
This makes it easy to plot shapes and lines in a two-dimensional space, allowing for visual representation and analysis:
  • The x-axis is the horizontal line where \(x\) values lie.
  • The y-axis is the vertical line where \(y\) values reside.
  • The origin, denoted as \(0,0\), is the point where these two axes intersect.
This system is foundational in mathematics for plotting equations and solving them graphically. For circles, understanding the Cartesian system helps us express the circle in terms of its standard equation, establishing its position and size relative to the origin.
Standard Form of a Circle
A circle in the Cartesian coordinate system is best understood using its standard form equation: \((x-h)^2+(y-k)^2=r^2\).
This formula is crucial as it allows us to clearly define the circle's position and measure its size. Here鈥檚 what each component signifies:
  • \(h\) and \(k\) are the x and y coordinates of the circle's center, respectively.
  • \(r\) represents the radius of the circle.
Applying this format, let's explore how it operates:
  • If you know the center of a circle is at (-9, 9) and the radius is 16, you can immediately plug these into the standard form.
  • This yields \((x + 9)^2 + (y - 9)^2 = 256\).
This equation reveals where the circle is located and how big it is, making it an essential tool for graphing and analyzing circles in a mathematical context.
Radius of a Circle
The radius of a circle is a key characteristic that defines its extent. It is the distance from the center of the circle to any point on its perimeter.
In the equation of a circle, the radius is represented by \(r\).
  • The radius is always a positive number and significantly contributes to the size of the circle.
  • In the aforementioned equation \((x-h)^2 + (y-k)^2 = r^2\), the radius is squared, turning the equation into \(r^2\) form.
When given a specific radius, as in this case where the radius is 16, we apply it like this:
  • Calculate the square of the radius, getting \(16^2 = 256\).
  • Incorporate this result into the circle's equation: \((x + 9)^2 + (y - 9)^2 = 256\).
Understanding and calculating the radius effectively enables us to figure out the circle's dimensions and contribute directly to solving its equation in practical problems.

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