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Chapter 13: Problem 16

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

Short Answer

Expert verified
Center: (0, 0); Radius: 4

Step by step solution

01

Identify Circle Equation

The given equation is in the form of a circle: \( x^2 + y^2 = 16 \). This is a standard form of a circle equation centered at the origin when the equation is \( x^2 + y^2 = r^2 \).
02

Determine Center of the Circle

The standard circle equation \( (x - h)^2 + (y - k)^2 = r^2 \), tells us that the center of the circle \((h, k)\) is at the origin when no \(h\) or \(k\) terms are added/subtracted from \(x\) or \(y\). Thus, for \( x^2 + y^2 = 16 \), the center is at \((0, 0)\).
03

Calculate the Radius of the Circle

In the standard equation \( (x - h)^2 + (y - k)^2 = r^2 \), the term \(r^2\) corresponds to the number on the other side of the equation. Here, \(r^2 = 16\), so \(r = \sqrt{16} = 4\). Thus, the radius is 4.
04

Prepare to Graph the Circle

To graph the circle, begin by plotting the center at \((0, 0)\) on the coordinate plane. Use the radius of 4 to measure 4 units in all directions from the center, marking points on the circle's perimeter.
05

Draw the Circle

Using the center at \((0, 0)\) and radius of 4, draw a smooth, round circle that extends evenly 4 units up, down, left, and right from the center point.

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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 crucial aspect of its geometry, serving as the fixed point equidistant from all points on the circle's edge. In the context of the standard form of a circle equation, which is \[(x - h)^2 + (y - k)^2 = r^2,\] the center is represented by the coordinates \((h, k)\).
  • If both \(h\) and \(k\) are zero, meaning the equation simplifies to \[x^2 + y^2 = r^2,\] the center of the circle is located at the origin \((0, 0)\).
  • Altering \(h\) and \(k\) shifts the circle on the coordinate plane away from the origin to the point \((h, k)\).
Understanding where the center is helps in graphing the circle accurately and in analyzing its position in a coordinate system. Knowing the center allows you to build a proper framework for determining other parameters such as the radius and aiding in tasks like translating and rotating the figure.
Radius of a Circle
The radius of a circle is the distance from the center of the circle to any point on its circumference. In the equation \[(x - h)^2 + (y - k)^2 = r^2,\] the term \(r^2\) signifies the square of the radius.
  • To find the radius, simply take the square root of this number, \(r = \sqrt{r^2}\).
  • This value is crucial, as it determines the size of the circle and is constant for any circle with a fixed equation.
In our example, given the equation \[x^2 + y^2 = 16,\] we can see that \(r^2 = 16\), so by calculating \(r = \sqrt{16} = 4\), the radius is fixed at 4 units. This tells us how far the circle extends from its center, allowing for accurate scaling and graph representations.
Standard Form of a Circle Equation
The standard form of a circle equation is a valuable tool in geometry, as it defines the circle in a coordinate plane clearly and concisely. The typical form is written as \[(x - h)^2 + (y - k)^2 = r^2,\] where \((h, k)\) represents the circle's center and \(r\) the radius.
  • This formulation provides an easy way to identify the circle's geometric properties, such as its center and radius, directly from the equation.
  • It facilitates tasks like graphing and applying transformations such as translations and rotations.
For example, in the equation \[x^2 + y^2 = 16,\] which is a specific form where \((h, k)\) are both zero, the circle is centered at the origin with \(r = 4\).This format contributes to understanding geometry's broader aspects, helping students and professionals recognize patterns and solve problems efficiently.

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Most popular questions from this chapter

Write each equation in standard form, if it is not alreacty so, and graph it. If the graph is a circle, give the coordinates of its center and its radius. If the graph is a parabola, give the coordinates of its vertex. $$ x^{2}+y^{2}+2 x-8=0 $$

Solve each system of equations for real values of x and y. $$ \left\\{\begin{array}{l} x-y=-1 \\ y^{2}-4 x=0 \end{array}\right. $$

Solve each system of equations for real values of x and y. $$ \left\\{\begin{array}{l} x^{2}=4-y \\ y=x^{2}+2 \end{array}\right. $$

Find each product. $$ \left(2 a^{-2}-b^{-2}\right)\left(2 a^{-2}+b^{-2}\right) $$

Solve each equation. $$ \frac{\log (8 x-7)}{\log x}=2 $$
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