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Chapter 1: Problem 33

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

Short Answer

Expert verified
The circle has center \( (0, 0) \) and radius \( 4 \).

Step by step solution

01

Identify the Standard Form

The given equation is in the form of a standard circle equation which is \(x^2 + y^2 = r^2\). In this form, \(r\) represents the radius and the center is at the origin \((h, k) = (0, 0)\).
02

Determine the Radius

Compare the given equation \(x^2 + y^2 = 16\) to the standard circle equation \(x^2 + y^2 = r^2\). Here, \(r^2 = 16\). Solving for \(r\), we get \(r = \sqrt{16} = 4\).
03

Write the Circle's Center

The center of the circle is determined by the equation \(x^2 + y^2 = r^2\). Since there are no \(h\) and \(k\) terms, the center is \( (h, k) = (0, 0) \).
04

Sketch the Graph

Draw the xy-plane. Mark the center of the circle at the origin \( (0, 0) \). From the center, measure a radius of \(4\) units in all directions to sketch the circle.

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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 Equation
To understand circles in geometry, it's essential to get familiar with their standard equation form. The standard form of a circle's equation is given by \[ (x - h)^2 + (y - k)^2 = r^2 \]. This formula tells us a lot about the circle:
  • \( (h, k) \) represents the center of the circle.
  • \( r \) is the radius.
In simpler terms, any circle can be described if we know its center and radius. If the equation of a circle looks like \[ x^2 + y^2 = r^2 \], then the center is at the origin \( (0, 0) \). In our given exercise, the equation \[ x^2 + y^2 = 16 \] matches this form, which means the center of the circle is at the origin (0, 0). Understanding this form helps ease solving and graphing circles.
Radius Calculation
Calculating the radius of a circle from its equation is straightforward when using the standard form. As previously mentioned, the equation for a circle in standard form is \[ (x - h)^2 + (y - k)^2 = r^2 \]. Here, \( r^2 \) represents the square of the radius.
To determine the radius, we need to compare the given equation of the circle to this standard form. For our exercise, the given equation is \[ x^2 + y^2 = 16 \]. Comparing it, we find that \[ r^2 = 16 \]. To find the actual radius, we take the square root of 16, giving us \[ r = \sqrt{16} = 4 \]. Thus, the radius of the circle is 4 units. This process is repeatable for any circle equation in standard form.
Graphing Circles
Graphing a circle can be enjoyable once you understand its equation and radius. Follow these steps to graph a circle successfully:
  • First, identify the center of the circle. In our given equation \[ x^2 + y^2 = 16 \], we determined the center to be at the origin \((0, 0)\).
  • Next, determine the radius. From our previous section, we calculated the radius to be 4 units.
  • To sketch the graph, begin by drawing your coordinate axes (xy-plane).
  • Mark the center of the circle at \((0, 0)\).
  • Measure 4 units outward from the center in all directions (up, down, left, and right).
  • Draw a curve that connects these points to form a circle.
Now, you have a visual representation of the circle with a center at \((0, 0)\) and a radius of 4. Regular practice of these steps makes graphing circles simpler and more intuitive.

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Most popular questions from this chapter

A pharmacist has 200 milliliters of a solution that is \(40 \%\) active ingredient. How much pure water should she add to the solution to get a solution that is \(25 \%\) active ingredient?

Solve \(x^{2}+2 x=9\).

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Find the distance between the points \((-3,5)\) and \((2,8)\) and the midpoint of the line segment joining the points.

$$\text { Solve } 3-5|x-4|=0$$
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