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Chapter 10: Problem 31

Use the center and the radius to graph each circle. $$ x^{2}+y^{2}=9 $$

Short Answer

Expert verified
In this case, the circle will be centered at the origin (0,0) and has a radius of 3 units.

Step by step solution

01

Identify the Center

Firstly, identify the center of the circle. Since there are no constants added to or subtracted from \(x^{2}\) and \(y^{2}\), the center of the circle is (0,0).
02

Calculate the Radius

Secondly, determine the radius. The right-hand side of the equation provides us with \(r^{2}\), the square of radius. Therefore, the radius is the square root of the right-hand side which is \(\sqrt{9} = 3\).
03

Graph the Circle

Lastly, plot the center and draw a circle with the previously calculated radius. Start with a dot at the origin (0,0) to represent the center. Then draw a circle of radius 3 units around this dot. The graph will be a circle centered at the origin with a radius of 3 units.

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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 a vital component in the equation of a circle. It acts as the point from which all points on the circle are equidistant. To find the center, look at the equation of the circle, typically given in the standard form \[(x - h)^2 + (y - k)^2 = r^2\]where
  • \( (h, k) \) represents the center of the circle.
  • \( r \) is the radius.
In cases where there are no additional terms added to or subtracted from \(x\) and \(y\), like in the equation \(x^2 + y^2 = 9\),
the center becomes the origin \((0, 0)\). This means that the circle is centered right where the coordinate axes intersect.
Understanding the center is key in graphing circles, as it determines the placement of the circle on the plane.
Radius of a Circle
The radius of a circle is the distance from the center to any point on the circle. In mathematical terms, it is derived directly from the equation of the circle. In the equation \[(x - h)^2 + (y - k)^2 = r^2\], \( r^2 \) is given on the right-hand side. To find the radius \( r \), simply take the square root of \( r^2 \).
For the equation \( x^2 + y^2 = 9 \), the radius can be found by evaluating \( \sqrt{9} = 3 \).
This tells us:
  • The distance from the center \((0, 0)\) to any point on the circle is 3 units.
When graphing, the radius helps define how large or small the circle will appear on the graph. Always remember, a longer radius results in a larger circle.
Equation of a Circle
The equation of a circle provides a complete description of all its fundamental properties and its placement on the graph. Normally written as \[(x - h)^2 + (y - k)^2 = r^2\],this equation lets us determine both the center \((h,k)\) and the radius \( r \) of the circle.
However, when the equation appears as \(x^2 + y^2 = r^2\), it signifies a special case: a circle centered at the origin \((0,0)\). Here, the work simplifies, since there are no linear terms to shift the center.
By understanding this equation, you can
  • Effortlessly identify whether a circle is at the origin
  • Determine its size by calculating \( r \)
The equation is fundamental to graphing, as it provides all necessary data to accurately draw the circle on a coordinate plane.

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Most popular questions from this chapter

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