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Chapter 13: Problem 12

Find the center and radius of each circle. Then graph the circle. $$ x^{2}+y^{2}=1 $$

Short Answer

Expert verified
The center is \(0,0\) and the radius is 1.

Step by step solution

01

Identify the Standard Form

The equation given is \(x^{2}+y^{2}=1\). This is a equation of a circle in its standard form \(x^{2} + y^{2} = r^{2}\).
02

Find the Radius

Compare \(x^{2} + y^{2} = 1\) with the standard form \(x^{2} + y^{2} = r^{2}\). Here, we can see that \(r^{2} = 1\), so the radius \(r = \sqrt{1} = 1\).
03

Find the Center

In the equation \(x^{2} + y^{2} = 1\), the center \(h, k\) is \(0, 0\) because there is no \(h\) and \(k\) term.
04

Graph the Circle

To graph the circle, plot the center at \(0, 0\) and draw a circle with a radius of 1 unit.

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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 helps easily identify the circle's center and radius. The general formula is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Here, \((h, k)\) represents the center of the circle, and \[r\] represents the radius. This form makes it straightforward to see the circle's geometric properties.
In the exercise, the given equation is:
\[ x^2 + y^2 = 1 \]
Notice it matches the standard form, with \((h, k)\) being \((0, 0)\) and \[r^2 = 1\]. This means the center of the circle is at the origin, and the radius squared is 1.
Radius Calculation
Determining the radius is key for understanding the size of the circle. Using the equation in standard form: \((x - h)^2 + (y - k)^2 = r^2\), it鈥檚 clear that \[r^2\] is the value on the right-hand side. Follow these steps to find the radius:
  • Compare the circle's equation with the standard form.
  • Identify the right-hand value as \[r^2\].
  • Solve for \[r\] by taking the square root of \[r^2\].

In our exercise:
\[ x^2 + y^2 = 1 \]
The value of \[r^2\] is 1. By taking the square root, \[ r = \sqrt{1} = 1 \]. Hence, the circle's radius is 1 unit.
Center of a Circle
To locate the center of the circle, look at the standard form equation: \[(x - h)^2 + (y - k)^2 = r^2 \]. Here, \(h, k\) are the coordinates of the center. If there are no \(h\) and \(k\) terms, the center is \(0, 0\).
In the given equation:
\[ x^2 + y^2 = 1 \]
There are no additional \(h\) or \(k\) terms added to \(x\) or \(y\). This means the center, \(h, k\), is \(0, 0\). So, plot the center at the origin on a graph.
Graphing Circles
Once you have the center and radius, graphing the circle is simple. Follow these steps to graph a circle:
  • Plot the center at \(h, k\).
  • From the center, measure the radius outward in all directions.
  • Draw a curve connecting these points.

For our specific exercise:
The center is at \[ (0, 0) \] and the radius is 1. To graph:
1. Plot \(0,0\).
2. Move 1 unit up, down, left, and right from the center.
3. Connect these points to form a circle.
This visual helps confirm the circle's properties and solve related problems easily.

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