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Chapter 10: Problem 11

Write an equation for each circle described below. $$\text { center at }(1,-4), r=\sqrt{17}$$

Short Answer

Expert verified
The equation of the circle is \((x - 1)^2 + (y + 4)^2 = 17\).

Step by step solution

01

Understand the Circle's Equation

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

Identify Values for the Equation

Given the center \((h, k)\) is \((1, -4)\) and the radius \(r\) is \(\sqrt{17}\). Identify \(h = 1\), \(k = -4\), and \(r = \sqrt{17}\).
03

Substitute Values into the Equation

Substitute \(h = 1\), \(k = -4\), and \(r = \sqrt{17}\) into the equation \((x - h)^2 + (y - k)^2 = r^2\). This gives the equation: \((x - 1)^2 + (y + 4)^2 = (\sqrt{17})^2\).
04

Simplify the Equation

Calculate \((\sqrt{17})^2\) which equals 17. So the equation becomes \((x - 1)^2 + (y + 4)^2 = 17\).

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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 one of the two main defining features of a circle. It is represented as the point \(h, k\) in the coordinate plane. This point is the same distance from every point on the circle. Think of it as the "heart" of the circle, regulating its position in space. If you know the center, you know exactly where the circle sits.

Determining the center is crucial for forming the circle's equation. From our exercise, the center given is at (1,-4).

  • The first value, 1, represents the x-coordinate (h) of the center.
  • The second value, -4, is the y-coordinate (k).
These coordinates help locate the circle precisely on the plane.
Radius of a Circle
The radius of a circle is equally critical, symbolizing the constant distance from the center to any point on the circle. It's usually represented by the letter \(r\). Picture the radius like a spoke of a wheel that reaches outward from the center to touch the edge.

In our problem, the radius is given as \(\sqrt{17}\).

Here are some useful points to remember about the radius of a circle:
  • It will always be a positive number since distance can't be negative.
  • Even though the radius here is \(\sqrt{17}\), once squared in the equation, it simplifies to 17, thus making calculations more straightforward.
This piece of information is vital because it determines the size of the circle.
Equation of a Circle
The equation of a circle is a mathematical representation that illustrates the circle's size and position. Written in the form \(x - h)^2 + (y - k)^2 = r^2\), it combines both the center and radius.

The equation shows everything about the circle in one concise format. Here's what each part corresponds to:
  • \(h\) and \(k\) are the center's coordinates. In the exercise, these are 1 and -4, respectively.
  • \(r\) is the radius of the circle. It's given as \(\sqrt{17}\), but its squared form (17) appears in the equation.
The equation created, \(x - 1)^2 + (y + 4)^2 = 17\), places this circle within the coordinate plane, outlining exactly where it is and its size. This formula remains consistent, showing us that no matter the circle's size or position, it can be read and understood using this simple equation structure.

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Most popular questions from this chapter

A paper company ships reams of paper in a box that weighs 1.3 pounds. Each ream of paper weighs 4.4 pounds, and a box can carry no more than 12 reams of paper. Which inequality best describes the total weight in pounds \(w\) to be shipped in terms of the number of reams of paper \(r\) in each box? \(\mathbf{F} \quad w \geq 1.3+4.4 r, r \geq 12\) \(\mathbf{G} \quad w=1.3+4.4 r, r \leq 12\) \(\mathbf{H} \quad w \leq 1.3+4.4 r, r \leq 12\) \(\mathrm{J} \quad \quad w=1.3+4.4 r, r \geq 12\)

Use the following information for Exercises 54 and 55 . Triangle \(A B C\) has vertices \(A(-3,2), B(4,-1),\) and \(C(0,-4)\) What are the coordinates of the image of \(\triangle A B C\) after a reflection in the \(y \text { -axis? (lesson } 9.1)\)

Determine whether each statement is sometimes, always, or never true. The central angle of a minor arc is an acute angle.

Compare and contrast an inscribed angle and a central angle that intercepts the same arc.

The radius, diameter, or circumference of a circle is given. Find the missing measures to the nearest hundredth. $$d=13, r=\underline{?}, C=\underline{?}$$.
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