Multiple Choice
Graph
15. Polar Equations
Graphing Other Common Polar Equations
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Identify whether the given equation is that of a cardioid, limaçon, rose, or lemniscate.
A
Cardioid
B
Limacon
C
Rose
D
Lemniscate
Verified step by step guidance1
Recognize that the given equation is in the form \( r^2 = -25 \cos 2\theta \). This is a key indicator of a lemniscate, which typically has the form \( r^2 = a^2 \cos 2\theta \) or \( r^2 = a^2 \sin 2\theta \).
Understand that a lemniscate is a type of polar graph that resembles a figure-eight or infinity symbol. The presence of \( \cos 2\theta \) in the equation suggests symmetry about the polar axis.
Compare the given equation to the standard forms of other polar graphs: a cardioid has the form \( r = a + a \cos \theta \) or \( r = a + a \sin \theta \), a limaçon has the form \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \), and a rose curve has the form \( r = a \cos n\theta \) or \( r = a \sin n\theta \). None of these match the given equation.
Note that the negative sign in \( -25 \cos 2\theta \) does not affect the classification as a lemniscate, but it does affect the orientation and position of the graph.
Conclude that the equation \( r^2 = -25 \cos 2\theta \) is indeed that of a lemniscate, as it fits the form \( r^2 = a^2 \cos 2\theta \) with \( a^2 = 25 \).
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