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Chapter 1: Problem 64

Find the center and radius of the circle. $$ (x+1)^{2}+(y-1)^{2}=16 $$

Short Answer

Expert verified
Center: \((-1, 1)\), Radius: 4.

Step by step solution

01

Identify the Circle Equation Form

The given equation \((x+1)^2 + (y-1)^2 = 16\) is in the standard form of a circle equation \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
02

Determine the Center

From the equation \((x+1)^2 + (y-1)^2 = 16\), compare it with \((x-h)^2 + (y-k)^2 = r^2\). Here, \(h = -1\) and \(k = 1\), so the center of the circle is \((-1, 1)\).
03

Calculate the Radius

The given equation \((x+1)^2 + (y-1)^2 = 16\) shows \(r^2 = 16\). To find \(r\), calculate \(r = \sqrt{16}\), which simplifies to \(r = 4\).
04

Summary

The center of the circle is \((-1, 1)\) and the radius is 4.

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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
The standard form of a circle's equation is crucial to understand when working with circles in coordinate geometry. This equation is written as:
  • \((x-h)^2 + (y-k)^2 = r^2\)
where:
  • \((h, k)\) represents the center of the circle
  • \(r\) is the radius of the circle.
In this format, all the essential properties of the circle are easily identifiable, such as the location of its center and its size through the radius.

This form is particularly helpful because it allows us to identify these properties at a glance, just by comparing the given equation to the standard form, making it a powerful tool in geometry problems.
Center of a Circle
Understanding the center of a circle is key when analyzing circle equations. When given the standard form of a circle \((x-h)^2 + (y-k)^2 = r^2\), the center is extracted as follows:
  • The term \( (x-h) \) reveals the x-coordinate of the center as \( h\).
  • The term \( (y-k) \) reveals the y-coordinate of the center as \( k \).
Therefore, the center of the circle is the point \((h, k)\). For instance, in the equation \((x+1)^2 + (y-1)^2 = 16\), the center can be found by recognizing that:
  • \(h = -1\)
  • \(k = 1\)
Thus, the center of the circle is at the coordinates \((-1, 1)\).

Remember, the signs of \(h\) and \(k\) are opposite in the equation compared to their values in the center, which is a common point of confusion that needs special attention.
Radius of a Circle
The radius is one of the fundamental aspects of a circle, signifying the distance from the center to any point on its circumference. In the equation's standard form, the radius is denoted as \(r\). From the form \((x-h)^2 + (y-k)^2 = r^2\), we see that:
  • \(r^2\) represents the square of the radius.
Thus, to find the radius \(r\), you take the square root of \(r^2\). For example, in the provided equation \((x+1)^2 + (y-1)^2 = 16\), \(r^2 = 16\).

Solving for \(r\), we calculate:
  • \(r = \sqrt{16}\), which simplifies to \(r = 4\).
This means the circle has a radius of 4. Understanding this process ensures you can quickly determine the size of any circle given in standard form.

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Most popular questions from this chapter

The amount of federal debt changed dramatically during the 30 years from 1970 to 2000 . (Sources: Department of the Treasury, Bureau of the Census.) A. In 1970 the population of the United States was \(203,000,000\) and the federal debt was \(\$ 370\) billion. Find the debt per person. B. In 2000 the population of the United States was approximately \(281,000,000\) and the federal debt was \(\$ 5.54\) trillion. Find the debt per person.

(Refer to Example 3.) Use the given \(f(x)\) to complete the following. (a) Calculate the average rate of change of \(f\) from \(x=1\) to \(x=2\) (b) Illustrate your result from part (a) graphically. $$ f(x)=x^{2} $$

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Air Temperature (Refer to Example 6 .) When the relative humidity is \(100 \%,\) air cools \(5.8^{\circ} \mathrm{F}\) for every 1 -mile increase in altitude. Give verbal, symbolic, graphical, and numerical representations of a function \(f\) that computes this change in temperature for an increase in altitude of \(x\) miles for \(0 \leq x \leq 3\).

Write a symbolic representation (formula) for a function \(g\) that calculates the given quantify. Then evaluate \(g(10)\) and interpret the result. The number of dollars in \(x\) quarters
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