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Chapter 1: Problem 84

In Exercises \(83-85,\) use a graphing utility to graph each circle whose equation is given. Use a square setting for the viewing window. $$x^{2}+10 x+y^{2}-4 y-20=0$$

Short Answer

Expert verified
The graph of the equation represents a circle with center at (-5, 2) and radius 7. It should be graphed using a square viewing window to ensure it appears correctly.

Step by step solution

01

Rewrite the Equation in Standard Form

The equation given is \(x^{2}+10 x+y^{2}-4 y-20=0\). Group together the x terms and y terms. The equation can be rewritten as \((x^2 + 10x) + (y^2 - 4y) = 20\). Then, add and subtract within parentheses to complete the square: \((x^2 + 10x + 25) - 25 + (y^2 - 4y + 4) - 4 = 20\). This simplifies to \((x + 5)^2 -25 + (y-2)^2 - 4 = 20\). So finally, it's \((x + 5)^2 + (y - 2)^2 = 49\).
02

Identify the Center and Radius

The equation \((x + 5)^2 + (y - 2)^2 = 49\) is in the standard form for the equation of a circle. The center of the circle is at the point (-5, 2) and the radius is \(\sqrt{49} = 7\).
03

Graph the Circle using a Graphing Utility

Using a graphing utility, input the equation \((x + 5)^2 + (y - 2)^2 = 49\). Remember to adjust the viewing window to a square setting, ensuring that the x and y axes are on the same scale, so the circle appears round, and centred at the point (-5, 2) with a radius of 7. Experiment with the plot window size to best fit the circle on the graph.

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Most popular questions from this chapter

My graph of \((x-2)^{2}+(y+1)^{2}=16\) is my graph of \(x^{2}+y^{2}=16\) translated two units right and one unit down.

The length of a rectangle exceeds the width by 13 yards. If the perimeter of the rectangle is 82 yards, what are its dimensions? (Section P.8, Example 6)

Solve for \(y: \quad x=\frac{5}{y}+4\)

In Exercises \(67-70,\) graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$\begin{aligned} x^{2}+y^{2} &=16 \\ x-y &=4 \end{aligned}$$

In Exercises \(53-64,\) complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}+8 x+4 y+16=0$$
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