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Chapter 7: Problem 16

Sketch the graph of the inequality.$$(x-1)^{2}+(y-4)^{2}>9$$

Short Answer

Expert verified
The graph of the inequality \((x-1)^2 + (y-4)^2 > 9\) is all the points in the xy-plane that are exterior to the dashed circle (not including the circle) with center at (1,4) and radius of 3 units.

Step by step solution

01

Identify the Form of Equation

Note that this inequality is in the standard form of the equation of a circle, \((x-a)^2 + (y-b)^2 = r^2\), where (a, b) is the center and r is the radius. In this case, the center is at (1, 4) and the radius is \(\sqrt{9} = 3\). However, here the inequality is '>', which means we are interested in the points outside of the circle.
02

Sketch the Circle

First, draw a circle with center at (1, 4) and radius of 3 units. It is formed by points where \((x-1)^2 + (y-4)^2 = 9\).
03

Shade the Exterior

The inequality \((x-1)^2 + (y-4)^2 > 9\) represents the region outside the circle. So all points in the plane that lie outside this circle are solutions to the inequality. We can show the solution set by shading the exterior of the circle.
04

Identify Boundary Circle

The circle \((x-1)^2 + (y-4)^2 = 9\) is the boundary of the solution set. As the inequality does not contain 'equals to' (=) symbol, the points on the circle are not included in the solution. To show this, the boundary circle should be drawn with a dashed line instead of a solid line.

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Key Concepts

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

Circle Equation
A circle equation serves as a fundamental concept in coordinate geometry. It is defined in its standard form as \[(x-a)^2 + (y-b)^2 = r^2\]where:
  • \( (a, b) \) represents the center of the circle.
  • \( r \) is the radius of the circle.
In the original exercise, the equation \[(x-1)^2 + (y-4)^2 = 9\]highlights a circle centered at \((1, 4)\) with a radius \( r = \sqrt{9} = 3 \). To convert an equation into its inequality form, a symbol such as '>' or '<' is used. This indicates a region relative to the circle, such as either inside or outside it.
Understanding how to adapt the circle equation to inequalities will help us determine which part of the coordinate plane represents the solution region for specific problems.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, allows us to study geometric figures using a coordinate system. In this system, every point is defined by an (x, y) pair, making it easier to visualize shapes such as circles.

For the original problem, plotting involves understanding how the center \((1, 4)\) and radius \(3\) affect the layout in the coordinate plane. The process involves:
  • Marking a point at \((1, 4)\).
  • Sketching a circle that extends with a radius of 3. This distance is measured from the center.
By leveraging these components, anyone can place geometric figures precisely within a graph. Recognizing how to apply transformations such as translations or reflections enhances our ability to graph complex shapes accurately.
Inequality Solution Sets
Inequality solution sets in the context of geometry reveal regions that either contain or exclude a circle. The inequality\[(x-1)^2 + (y-4)^2 > 9\]specifically describes the space outside the described circle.


When solving this inequality:
  • A circle is drawn as the baseline using \((x-1)^2 + (y-4)^2 = 9\).
  • Because the inequality symbol is '>', the solution set includes points outside this circle.
  • The boundary for this region is indicated by using a dashed line for the circle. This shows that the boundary itself is not included.
Inequality symbols '<' or '>' mark whether solutions are inside or outside the curve. Understanding how to depict these regions on a graph is essential for illustrating solutions to inequality problems accurately.

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Most popular questions from this chapter

Find values of \(x, y,\) and \(\lambda\) that satisfy the system. These systems arise in certain optimization problems in calculus, and \(\lambda\) is called a Lagrange multiplier. $$\left\\{\begin{aligned} 2+2 y+2 \lambda &=0 \\ 2 x+1+\lambda &=0 \\ 2 x+y-100 &=0 \end{aligned}\right.$$

Solve the system of linear equations and check any solutions algebraically. $$\left\\{\begin{array}{l} 2 x+y+3 z=1 \\ 2 x+6 y+8 z=3 \\ 6 x+8 y+18 z=5 \end{array}\right.$$

Write the partial fraction decomposition of the rational expression. Then assign a value to the constant \(a\) to check the result algebraically and graphically. $$\frac{1}{(x+1)(a-x)}$$

The linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the minimum and maximum values of the objective function (if possible) and where they occur. Objective function: \(z=x+y\) Constraints: $$\begin{aligned}x & \geq 0 \\\y & \geq 0 \\ -x+y & \leq 0 \\\\-3 x+y & \geq 3\end{aligned}$$

Find the equation of the circle $$x^{2}+y^{2}+D x+E y+F=0$$ that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle. $$(0,0),(0,6),(3,3)$$
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