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Chapter 10: Problem 51

Find the center and the radius of each circle. Then graph the circle. $$x^{2}+y^{2}=8$$

Short Answer

Expert verified
Center: (0,0); Radius: \(the square root of 8\)

Step by step solution

01

Identify the Equation Form

Recognize that the equation of the circle is in standard form, which is \(x^{2} + y^{2} = r^{2}\).
02

Determine the Center

Since the equation \(x^{2} + y^{2} = 8\) does not have any terms with \(x\) or \(y\) alone, the center of the circle is at the origin \( (0,0)\).
03

Calculate the Radius

The equation is in the form \(x^{2} + y^{2} = r^{2}\). Match \(r^{2} = 8\). Therefore, the radius is \(\frac{\text{the square root of 8}}{2}\), which simplifies to \(2\frac{\text{the square root of 2}}{1}\).
04

Graph the Circle

To graph the circle, plot the center at (0,0). Draw a circle around the center with a radius of \(2\frac{\text{the square root of 2}}{1}\).

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Key Concepts

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

Center of a circle
Understanding the center of a circle is fundamental. In this particular exercise, we observe the equation \(x^2 + y^2 = 8\). This is in the standard form of a circle's equation, which is \(x^2 + y^2 = r^2\). Notice, there are no terms with just \(x\) or \(y\). This omission tells us that the center of the circle is at the origin point \((0,0)\).

If there were additional terms such as \((x - h)^2 + (y - k)^2 = r^2\), the center would instead be at the point \((h,k)\). Always look for these terms to identify the circle's center easily.
Radius calculation
Calculating the radius of a circle is another crucial step. Given our circle's equation \(x^2 + y^2 = 8\), we can directly compare it to the standard form \(x^2 + y^2 = r^2\). Here, \(r^2 = 8\). To find the radius \(r\), we need to take the square root of both sides:

\[ r = \sqrt{8} \]

This simplifies to \[ r = 2\sqrt{2} \]

Therefore, the radius of the circle is \(2\sqrt{2}\). Remember: whenever you see an equation in the form of \(r^2\), simply find the square root to determine your radius.
Graphing circles
Graphing a circle involves a few simple steps. Firstly, identify the circle's center and radius, as we did earlier. With our circle centered at \((0,0)\) and radius \(2\sqrt{2}\), we move to graphing.

  • Start by plotting the center point at \((0,0)\).
  • Using a compass or by measuring, draw a circle around this center point, ensuring it reaches out to the radius length of \(2\sqrt{2}\) in all directions.


This symmetrical shape, extending evenly from the center, completes your circle. Practicing these steps with different equations will make you proficient in graphing circles accurately and quickly.

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