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Chapter 0: Problem 53

Determine the center and radius of each circle and sketch the graph. $$ x^{2}+y^{2}=16 $$

Short Answer

Expert verified
The center of the circle is (0, 0) and the radius is 4.

Step by step solution

01

Identify the circle equation form

The general form of a circle equation is \[ (x - h)^2 + (y - k)^2 = r^2 \]. Our given equation is \[ x^2 + y^2 = 16 \].
02

Compare the given equation to the standard form

Notice that in the given equation \[ x^2 + y^2 = 16 \], there are no \( h \) and \( k \) terms, which implies \( h = 0 \) and \( k = 0 \). Therefore, the center of the circle \((h, k)\) is \((0, 0)\).
03

Determine the radius

The number on the right side of the equation \[ 16 \] represents \( r^2 \). To find the radius \( r \), take the square root of 16: \[ r = \sqrt{16} = 4 \]. Therefore, the radius of the circle is 4.
04

Sketch the graph

To sketch the graph, plot the center of the circle at the origin \((0, 0)\). Then, draw a circle with a radius of 4 units around the origin.

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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
Understanding the standard form of a circle equation is essential. It鈥檚 given by: \( (x - h)^2 + (y - k)^2 = r^2 \) Here:
    \t
  • \( (h, k) \) are the coordinates of the center of the circle.
    • \t
  • \( r \) is the radius of the circle.
In the standard form, \( h \) and \( k \) shift the circle from the origin. For example, the equation \( (x - 2)^2 + (y + 3)^2 = 25 \) represents a circle with its center at \( (2, -3) \) and radius \( \sqrt{25} = 5 \). In the given problem, we have \( x^2 + y^2 = 16 \). This represents a circle centered at the origin because \( h = 0 \) and \( k = 0 \).
circle radius
The radius of a circle is the distance from its center to any point on the circle. In the standard form equation \( (x - h)^2 + (y - k)^2 = r^2 \), the radius \( r \) is the square root of the number on the right side of the equation. For the given equation \( x^2 + y^2 = 16 \):
    \t
  • First, recognize that the number \( 16 \) represents \( r^2 \).
    • \t
  • Then, take the square root of \( 16 \), which is \( 4 \).
This means the circle鈥檚 radius is \( 4 \). You can picture this as stretching 4 units from the center in every direction. Thus, if you start at the origin \( (0, 0) \), you go out 4 units to any point on the circle鈥檚 edge.
circle center
The center of a circle determines its position in the coordinate plane. In the standard form equation \( (x - h)^2 + (y - k)^2 = r^2 \), the center is given by the point \( (h, k) \). For the equation \( x^2 + y^2 = 16 \), there are no \( h \) and \( k \) terms, indicating that both \( h \) and \( k \) are 0. Therefore, the circle is centered at the origin \( (0, 0) \). This concept is crucial because the center point affects how you draw the circle and calculate distances or symmetry. Knowing the position of a circle鈥檚 center allows you to understand the circle鈥檚 placement and relationships with other geometrical figures in the plane.

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