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Chapter 1: Problem 49

Determine the center and radius of each circle and sketch the graph. See the rule for completing the square on page 31 $$x^{2}+y^{2}=9$$

Short Answer

Expert verified
The center is \((0,0)\) and the radius is \(3\).

Step by step solution

01

Identify the equation form

Observe the given equation is in the form of the standard circle equation \(x^2 + y^2 = 9\)
02

Compare with the standard form

The standard form of a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\), where \(h, k\) is the center and \(r\) is the radius.
03

Determine the center

By comparing \((x-h)^2 + (y-k)^2 = r^2\) with \(x^2 + y^2 = 9\), it can be seen that \(h = 0\) and \(k = 0\), so the center is \((0,0)\).
04

Determine the radius

Equate \(r^2\) to 9 from \((x-h)^2 + (y-k)^2 = r^2\). Therefore, \(r^2 = 9\) and taking the square root of both sides, \(r = 3\).
05

Conclusion

The center of the circle is \((0,0)\) and the radius is \(3\).

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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 is a way to write the equation of a circle so that you can easily identify its center and radius. The general formula is:
Completing the Square
Completing the square is a method used to convert a quadratic equation into a perfect square trinomial. This makes it easier to identify the key features of circles, such as their centers and radii. For example, if you have an equation like...
Radius and Center Determination
To determine the radius and center of a circle from its equation, you need to rewrite the equation in its standard form by completing the square if necessary. Here's how to do it step-by-step: ...

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