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Chapter 2: Problem 69

\(65-72\) . Show that the equation represents a circle, and find the center and radius of the circle. $$ x^{2}+y^{2}+x=0 $$

Short Answer

Expert verified
The circle has center \((-\frac{1}{2}, 0)\) and radius \(\frac{1}{2}\).

Step by step solution

01

Identify the Equation Type

We are given the equation \(x^2 + y^2 + x = 0\). This equation is of the form \(x^2 + y^2 + ax + by + c = 0\), which can represent a circle. To confirm, let's rewrite it in the standard form of a circle.
02

Complete the Square for 'x'

Rewrite the equation to group \(x\) terms together: \(x^2 + x + y^2 = 0\). Complete the square for \(x\) by taking half of the coefficient of \(x\) (which is \(1\)), squaring it to get \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\), and adjusting the equation:\[x^2 + x + \frac{1}{4} + y^2 = -\frac{1}{4}\].
03

Simplify the Expression

The expression \(x^2 + x + \frac{1}{4}\) can be rewritten as \(\left(x + \frac{1}{2}\right)^2\), turning the equation into:\[\(\left(x + \frac{1}{2}\right)^2 + y^2 = \frac{1}{4}\)\].
04

Identify the Center and Radius

The equation \(\left(x + \frac{1}{2}\right)^2 + y^2 = \frac{1}{4}\) is now in the standard form \((x - h)^2 + (y - k)^2 = r^2\).- The center \((h, k)\) of the circle is \((-\frac{1}{2}, 0)\).- The radius \(r\) is the square root of \(\frac{1}{4}\), which is \(\frac{1}{2}\).

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Key Concepts

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

Completing the Square
When transforming an equation into the standard form of a circle, completing the square is a helpful method. It helps to simplify quadratic expressions. Let鈥檚 break down the steps to complete the square:
  • First, focus on the terms involving one variable, such as the x-terms.
  • Take the coefficient of the x term, which is the number in front of the x"). If the term is x"), consider the coefficient to be 1 here.
  • Divide this coefficient by 2. For example, for the equation x^2 + x"): \(\frac{1}{2} = 0.5\).
  • Next, square that result: \(0.5^2 = 0.25\).
  • Add and subtract this square inside the equation to maintain balance.
  • Rewrite the expression as a binomial square. For example, x^2 + x + 0.25"): becomes \(\left(x + 0.5\right)^2\).
This method allows you to rearrange expressions neatly, setting the stage to see equations in their standard forms, particularly for determining circle equations.
Center of a Circle
The center of a circle is crucial for understanding its position in the coordinate plane. For a circle equation in the standard form \((x - h)^2 + (y - k)^2 = r^2\), the center is denoted by the coordinates \((h, k)\). Here鈥檚 how to find it from the equation through a simple process:
  • Look for transformations applied to the \(x\) and \(y\) terms.
  • For \(\left(x + a\right)^2\): the transformation is \(x - (-a)\), indicating a horizontal shift by \(-a\).
  • For \(\left(y + b\right)^2\): the transformation is \(y - (-b)\), showing a vertical shift by \(-b\).
  • Remember: If you see \(x + 0.5\), this corresponds to \(x - (-0.5)\).
In the given example equation, the terms \((x + \frac{1}{2})^2 \) and \(y^2\) indicate shifts. Thus, the center is at \((-\frac{1}{2}, 0)\). Think of these as adjustments that 'move' the circle from the origin to its actual center point.
Radius of a Circle
The radius of a circle tells us how far away the boundary is from the center. In the circle equation \((x - h)^2 + (y - k)^2 = r^2\), the radius is represented by r. To find it:
  • Identify the constant on the right side of the equation, which is \(r^2\).
  • Take the square root of this constant to find the radius \((r)\).
  • For instance, if the original equation is simplified to: \(\left(x + \frac{1}{2}\right)^2 + y^2 = \frac{1}{4}\), then \((r^2 = \frac{1}{4})\).
  • The square root of \(\frac{1}{4}\) is \(\frac{1}{2}\).
In our example, the given equation confirms the radius is \(\frac{1}{2}\). The radius provides valuable information about the circle鈥檚 scale and how it spans the coordinate plane.

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Most popular questions from this chapter

Radiation Energy The total radiation energy \(E\) emitted by a heated surface per unit area varies as the fourth power of its absolute temperature \(T\) . The temperature is 6000 \(\mathrm{K}\) at the surface of the sun and 300 \(\mathrm{K}\) at the surface of the earth. (a) How many times more radiation energy per unit area is produced by the sun than by the earth? (b) The radius of the earth is 3960 mi and the radius of the sun is \(435,000\) mi. How many times more total radiation does the sun emit than the earth?

Flea Market The manager of a weekend flea market knows from past experience that if she charges \(x\) dollars for a rental space at the flea market, then the number \(y\) of spaces she can rent is given by the equation \(y=200-4 x\) (a) Sketch a graph of this linear equation. (Remember that the rental charge per space and the number of spaces rented must both be nonnegative quantities.) (b) What do the slope, the \(y\) -intercept, and the \(x\) -intercept of the graph represent?

Kepler's Third Law Kepler's Third Law of planetary motion states that the square of the period \(T\) of a planet (the time it takes for the planet to make a complete revolution about the sun) is directly proportional to the cube of its average distance \(d\) from the sun. (a) Express Kepler's Third Law as an equation. (b) Find the constant of proportionality by using the fact that for our planet the period is about 365 days and the average distance is about 93 million miles. (c) The planet Neptune is about \(2.79 \times 10^{9}\) mi from the sun. Find the period of Neptune.

Use slopes to determine whether the given points are collinear (lie on a line). $$ \begin{array}{l}{\text { (a) }(1,1),(3,9),(6,21)} \\ {\text { (b) }(-1,3),(1,7),(4,15)}\end{array} $$

Manufacturing Cost The manager of a furniture factory finds that it costs \(\$ 2200\) to manufacture 100 chairs in one day and \(\$ 4800\) to produce 300 chairs in one day. (a) Assuming that the relationship between cost and the number of chairs produced is linear, find an equation that expresses this relationship. Then graph the equation. (b) What is the slope of the line in part (a), and what does it represent? (c) What is the \(y\) -intercept of this line, and what does it represent?
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