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Chapter 13: Problem 18

Find the center and radius of each circle and graph it. $$ (x+4)^{2}+y^{2}=1 $$

Short Answer

Expert verified
The circle's center is \((-4, 0)\), and its radius is 1.

Step by step solution

01

Identify the Standard Circle Equation

The standard form of a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
02

Compare with Given Equation and Determine Center

The given equation is \((x+4)^2 + y^2 = 1\). Compare this with the standard form: \((x - (-4))^2 + (y - 0)^2 = 1\) indicates that the center \((h, k)\) is \((-4, 0)\).
03

Determine the Radius

In the equation \((x+4)^2 + y^2 = 1\), the right side equals \(r^2\). Thus, \(r^2 = 1\), and therefore, \(r = \sqrt{1} = 1\).
04

Graph the Circle

To graph the circle, plot the center \((-4, 0)\) on the coordinate plane. From this center, use the radius of 1 to mark points that are 1 unit away in all directions (up, down, left, right) to shape the circle.

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Key Concepts

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

Standard Form of a Circle
In mathematics, understanding the equation of a circle is crucial for graphing and analyzing circular shapes. The standard form of a circle's equation provides a simple way to identify a circle's geometric properties directly from the equation. The standard form of a circle is written as \((x-h)^2 + (y-k)^2 = r^2\).
Here are the components involved:
  • \((h, k)\) represents the coordinates of the circle's center.
  • \(r\), the radius, is the distance from the center to any point on the circle.
This format makes it easy to pinpoint the circle's center and radius just by inspection. It's a powerful and intuitive way to bring a circle's algebraic equation to life in a geometrical form. Whenever you encounter a circle equation, rearrange it to match this standard format to make it easier to work with.
Center of a Circle
The center of a circle is a crucial point that lies exactly in the middle of the circle. It is denoted by the coordinate pair \((h, k)\) in the standard circle equation \((x-h)^2 + (y-k)^2 = r^2\).
Finding the center from a standard-form equation is straightforward.
Consider the equation \((x+4)^2 + y^2 = 1\). To find the center:
  • Identify \(h\) by looking at the transformation \((x-h)\). Here, \(x+4 = x - (-4)\), so \(h = -4\).
  • Identify \(k\) by looking at \((y-k)\). Here, \(y^2 = (y-0)^2\), so \(k = 0\).
Therefore, the center of the circle is \((-4, 0)\). The center is the pivotal point around which the circle is perfectly balanced, playing a key role in defining its shape and position on a graph.
Radius of a Circle
The radius of a circle is one of its most defining features. It is the constant distance from the circle's center to any point on its edge. Radii are always equal around the center, creating a uniform distance in all directions. In the circle equation \((x-h)^2 + (y-k)^2 = r^2\), \(r^2\) is the square of the radius.
To find the radius from an equation:
  • Look for the value on the equation's right side, which equals \(r^2\).
  • Take the square root of this value to find \(r\).
For example, in the equation \((x+4)^2 + y^2 = 1\), \(r^2 = 1\). Taking the square root, we find \(r = \sqrt{1} = 1\). The radius is not just a measurement but also an indicator of the circle's size, defining how far it stretches from its center.

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Most popular questions from this chapter

Use a nonlinear system of equations to solve each problem. Investing. Carol receives \(\$ 67.50\) annual income from one investment. John invested \(\$ 150\) more than Carol at an annual rate of \(1 \frac{1}{2} \%\) more. John's annual income is \(\$ 94.50 .\) What are the amount and rate of Carol's investment? (Hint: There are two answers.)

Solve each system. See Using Your Calculator: Solving Systems of Equations. $$ \left\\{\begin{array}{l} x^{2}-6 x-y=-5 \\ x^{2}-6 x+y=-5 \end{array}\right. $$

Write each equation in standard form, if it is not alreacty so, and graph it. If the graph is a circle, give the coordinates of its center and its radius. If the graph is a parabola, give the coordinates of its vertex. $$ x=-6(y-1)^{2}+3 $$

Write each equation in standard form, if it is not alreacty so, and graph it. If the graph is a circle, give the coordinates of its center and its radius. If the graph is a parabola, give the coordinates of its vertex. $$ x=\frac{1}{2} y^{2}+2 y $$

Find each product. $$3 x^{-2} y^{2}\left(4 x^{2}+3 y^{-2}\right)$$
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