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Chapter 11: Problem 37

(a) find the center and radius, then (b) graph each circle. $$ x^{2}+y^{2}=64 $$

Short Answer

Expert verified
The center is \((0, 0)\) and the radius is 8 units.

Step by step solution

01

Identify the equation

Notice the given equation of the circle: \(x^2 + y^2 = 64\). A standard circle equation is \((x - h)^2 + (y - k)^2 = r^2\), where \(h, k\) are the coordinates of the center, and \(r\) is the radius.
02

Compare with standard form

Compare the given equation \(x^2 + y^2 = 64\) with the standard form \((x - h)^2 + (y - k)^2 = r^2\). Observe that \(h = 0\) and \(k = 0\). This shows that the center of the circle is at the origin \((0, 0)\).
03

Determine the radius

The equation in the standard form reveals \(r^2 = 64\). Solving for the radius, we get \(r = \sqrt{64} = 8\). Hence, the radius of the circle is 8 units.
04

Graph the circle

To graph the circle, plot the center at \((0, 0)\) on the coordinate plane. Use a compass or draw a circle with radius 8 units around the center. Ensure the circle passes through the points \((8, 0)\), \((0, 8)\), \((-8, 0)\), and \((0, -8)\).

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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 equation simplifies understanding and calculations. It is written as \((x - h)^2 + (y - k)^2 = r^2 \), where \(h\) and \(k\) are the coordinates of the center of the circle, and \(r \) is the radius. This equation shows the precise location and size of the circle on the coordinate plane.
In the given exercise, the equation \((x^2 + y^2 = 64)\) can be directly compared to the standard form. Since there are no terms involving \(h \) or \(k \), it means both \(h \) and \(k \) are zero. Hence, the center of the circle is at the origin \((0, 0)\). Recognizing this form helps easily identify the circle's key features.
Radius of a Circle
The radius of a circle is the distance from the center to any point on the circle. To find the radius from the standard form, look at the right-hand side of the equation \((x - h)^2 + (y - k)^2 = r^2 \). The term \(r^2 \) gives the square of the radius.
In our exercise, we have \(x^2 + y^2 = 64 \). Here, \(64 \) represents \(r^2 \). To find \(r\), take the square root of \(64 \), which results in \(r = 8 \). Thus, the radius of the circle is 8 units. Knowing the radius is crucial for graphing and understanding the circle鈥檚 size.
Graphing Circles
Graphing circles on a coordinate plane can be simple if you know the center and radius. Start by plotting the center point. In our exercise, this is at the origin \((0, 0)\).
Next, measure the radius outward from the center. Since the radius is 8 units, mark points 8 units away in all directions: to the right \((8, 0)\), up \((0, 8)\), to the left \((-8, 0)\), and down \((0, -8)\).
Finally, draw the circle by connecting these points smoothly. You can use a compass or freehand if accuracy is not critical.
Graphing circles clearly shows their size and position, reinforcing the concepts learned from the equation.

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