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Chapter 13: Problem 64

Graph using a graphing calculator. $$x^{2}+y^{2}-10 x-11=0$$

Short Answer

Expert verified
The graph is a circle with center (5, 0) and radius 6.

Step by step solution

01

Identify the Equation Type

The given equation is in the form: \[ x^2 + y^2 - 10x - 11 = 0 \]. This represents a circle.
02

Complete the Square for x and y Terms

To convert this into the standard form of a circle, complete the square: \[(x^2 - 10x) + y^2 - 11 = 0\]. First, complete the square for the x-terms: \[ x^2 - 10x = (x - 5)^2 - 25 \].
03

Rewrite the Equation

Substitute back into the original equation: \[(x - 5)^2 - 25 + y^2 - 11 = 0\]. Combine constants on the left side: \[(x - 5)^2 + y^2 = 36\].
04

Identify the Circle's Center and Radius

The equation \[(x - 5)^2 + y^2 = 36\] is in the standard form \[(x - h)^2 + (y - k)^2 = r^2\]. Here, the center (h, k) = (5, 0) and the radius r = 6.
05

Input the Equation into the Graphing Calculator

Enter \[(x - 5)^2 + y^2 = 36\] into the graphing calculator. Use the circle tool or convert it to a function form if necessary.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Completing the Square
Completing the square is a method used to transform a quadratic equation into a perfect square trinomial.
This technique is especially helpful when dealing with the equation of a circle. To complete the square:
  • Group the x-terms together and do the same for y-terms if necessary.
  • Add and subtract the necessary constant to form a perfect square trinomial.
  • Rewrite the grouped terms as perfect square binomials.
Let's illustrate with the given equation: \(x^2 + y^2 - 10x - 11 = 0\).
Step 1: Focus on the x-terms: \(x^2 - 10x\)
Step 2: To complete the square, add and subtract \(25\) because \(25 = \left(\frac{-10}{2}\right)^2\).
Step 3: The grouping becomes \(x^2 - 10x = (x - 5)^2 - 25\).
This method allows us to convert the general form of a quadratic equation into a more manageable form, aiding in identifying centers and radii of circles.
Circle Equation
A circle's equation in standard form provides a clear way to identify the circle's center and radius. It's written as \((x-h)^2 + (y-k)^2 = r^2\), where \( (h, k) \) is the center and \( r \) is the radius. From the original equation \(x^2 + y^2 - 10x - 11 = 0\):
  • After completing the square, we get \( (x - 5)^2 + y^2 = 36 \).
  • The center is found by looking at the values inside the squared terms. Here, it is \( (5, 0) \).
  • The term 36 represents \( r^2 \), so the radius \( r = \sqrt{36} = 6 \).
The new form \((x - 5)^2 + y^2 = 36\) shows the circle is centered at \( (5, 0) \), with a radius of 6 units. This equation helps in graphing and visualizing the circle accurately.
Graphing Calculator
Graphing calculators are powerful tools for visualizing equations, including circles. Here's how to use one for the equation \( (x - 5)^2 + y^2 = 36 \):
  • Turn on the graphing calculator and enter the equation.
  • Ensure it's in the correct form, and use the circle tool if your calculator has one.
  • If needed, set up the graphing window to properly display the circle's center and radius.
  • For a better view, adjust the viewing window to include the center \( (5, 0) \) and extend to cover the radius of 6 units in all directions.
Using a graphing calculator helps you accurately plot the circle and understand its dimensions.
It's especially handy for checking your work and visualizing the structure of the circle equation you derived.

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