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Chapter 8: Problem 28

Convert the polar equation to rectangular form and sketch its graph. $$ r=-2 $$

Short Answer

Expert verified
The rectangular form of the given equation \(-2\), is \(x^2 + y^2 = 4\), which represents a circle with a radius of 2 centered at the origin. The graph is a circle reflected across the origin.

Step by step solution

01

Convert Polar Equation to a Rectangular Form

Recall that the polar coordinates are related to the rectangular coordinates through the formulas: \(x=r \cos(\theta)\) and \(y=r \sin(\theta)\). The given polar equation is \(r = -2\). Insert this into the equations for the conversion. Since \( r = -2 \), we can replace \( r \) in the conversion formulas directly to get \(x = -2 \cos(\theta)\) and \(y = -2 \sin(\theta)\).
02

Solving for the Rectangular Coordinate

Although we have equations for x and y, a single equation is needed that relates x and y to draw a graph in rectangular form. Since we know that \(x^2 + y^2 = r^2\) in polar coordinates, substitute \( r = -2 \) into this equation to get \(x^2 + y^2 = (-2)^2 = 4\). This is the equation of a circle centered at the origin with a radius of 2 in rectangular coordinates.
03

Sketch the Graph

Draw a circle centered at the origin with a radius of 2. Since the initial r-value given was negative, indicating a direction opposite from the usual direction, the actual graph would be the reflection of the circle across the origin.

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Most popular questions from this chapter

In Exercises \(27-38,\) find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.) \(\frac{\text { Conic }}{\text { Parabola }} \quad \frac{\text { Eccentricity }}{e=1} \quad \frac{\text { Directrix }}{x=1}\)

Eliminate the parameter and obtain the standard form of the rectangular equation. $$ \text { Circle: } x=h+r \cos \theta, \quad y=k+r \sin \theta $$

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=1+\frac{1}{t}, \quad y=t-1 $$

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ x=4 \sin 2 \theta, y=2 \cos 2 \theta $$

Identify each conic. (a) \(r=\frac{5}{1-2 \cos \theta}\) (b) \(r=\frac{5}{10-\sin \theta}\) (c) \(r=\frac{5}{3-3 \cos \theta}\) (d) \(r=\frac{5}{1-3 \sin (\theta-\pi / 4)}\)
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