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Chapter 0: Problem 48

State the center and radius of the circle with the given equations. $$(x+1)^{2}+(y+2)^{2}=8$$

Short Answer

Expert verified
The center is (-1, -2) and the radius is \(2\sqrt{2}\).

Step by step solution

01

Identify Standard Form of Circle Equation

The given equation is in the standard form of \((x-h)^2 + (y-k)^2 = r^2\), where \((h,k)\) is the center of the circle and \(r\) is the radius. In the given equation, \((x+1)^2+(y+2)^2=8\), by comparing, we identify that \(h = -1\), \(k = -2\), and \(r^2 = 8\).
02

Determine the Center of the Circle

Using the values identified from the standard form, the center \((h, k)\) of the circle is \((-1, -2)\).
03

Calculate the Radius of the Circle

The equation gives us \(r^2 = 8\). To find the radius \(r\), we take the square root of both sides, resulting in \(r = \sqrt{8} = 2\sqrt{2}\).

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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 a simple yet powerful way to describe a circle in a coordinate plane. It is given by the expression \[(x-h)^2 + (y-k)^2 = r^2\]where:
  • \( (h, k) \) represents the center of the circle.

  • \( r \) is the radius of the circle, with \( r^2 \) being on the right side of the equation.
This form is particularly useful because it directly reveals both the circle's center and its radius. By comparing any given circle equation to this standard form, you can easily extract these characteristics. For instance, consider an equation of the form \[(x+1)^2+(y+2)^2=8\].Here, you would adjust signs based on the standard form definition (\(-(x-h)^2\), etc.). This comparison helps identify necessary information about the circle.
Center of a Circle
The center of a circle is a crucial point which defines the location of the circle within the coordinate plane. In the standard form equation \[(x-h)^2 + (y-k)^2 = r^2\],the center is indicated by the point \( (h, k) \).When you are given a circle equation in the form \[(x+1)^2 + (y+2)^2 = 8\],you need to perform the following steps to determine the center:
  • Compare it to the standard form \((x-h)^2 + (y-k)^2 = r^2\).

  • Identify \(h\) and \(k\) by adjusting the signs from the equation to match \(-(x-h)\) and \(-(y-k)\).

  • In this example, \(h = -1\) and \(k = -2\). Therefore, the center is at the point \((-1, -2)\).
The center is the anchor point for the circle around which it is drawn, and understanding it helps in graphing the circle as well.
Radius of a Circle
The radius of a circle is the distance from its center to any point on the circumference. In the standard form \[(x-h)^2 + (y-k)^2 = r^2\],\( r \) is the radius.To find the radius from a given circle equation like \[(x+1)^2 + (y+2)^2 = 8\],follow these steps:
  • Note the right side of the equation. Here, it shows \(8\) which is \(r^2\).

  • Solve for \(r\) by taking the square root of \(r^2\). That means \( r = \sqrt{8} \).

  • Simplify the square root to get \( r = 2\sqrt{2} \).
The radius provides a measure of the size of the circle, showing how wide it stretches from the center to its edge. In graphical representations, this measurement is vital for drawing precise circles.

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Most popular questions from this chapter

Find the equation of the line that passes through the given point and also satisfies the additional piece of information. Express your answer in slope- intercept form, if possible. (1,4)\(;\) perpendicular to the line \(-\frac{2}{3} x+\frac{3}{2} y=-2\)

Mike's home phone plan charges a flat monthly fee plus a charge of \(\$ .05\) per minute for long-distance calls. The total monthly charge is represented by \(y=0.05 x+35\) \(x \geq 0,\) where \(y\) is the total monthly charge and \(x\) is the number of long-distance minutes used. Interpret the meaning of the \(y\) -intercept.

Explain the mistake that is made. Given the slope, classify the line as rising, falling, horizontal, or vertical. a. \(m=0\) b. \(m\) undefined c. \(m=2\) d. \(m=-1\) Solution: a. vertical line b. horizontal line c. rising d. falling These are incorrect. What mistakes were made?

Solve for the indicated variable in terms of other variables. Solve \(P=E I-R I^{2}\) for \(I\).

a. Solve the equation \(x^{2}-2 x=b, b=8\) by first writing in standard form. Now plot both sides of the equation in the same viewing screen \(\left(y_{1}=x^{2}-2 x \text { and } y_{2}=b \right)\). At what. \(x\) -values do these two graphs intersect? Do those points agree with the solution set you found? b. Repeat (a) for \(b=-3,-1,0,\) and 5
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