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Chapter 10: Problem 69

Points of Intersection Explain why finding points of intersection of polar graphs may require further analysis beyond solving two equations simultaneously.

Short Answer

Expert verified
The need for further analysis beyond solving equations when finding points of intersection in polar graphs boils down to the nature of polar coordinates. The same point can have different representations in polar format, with varying r and 胃. Therefore, all possible values of r and 胃 must be considered.

Step by step solution

01

Understanding Polar Coordinates

Firstly, to approach this problem, it's important to understand that polar coordinates are represented as (r, 胃), where r represents the radial distance from the origin and 胃 represents the counterclockwise angle from the positive x-axis. Different from Cartesian coordinates (x, y), a single point can be represented in many ways in polar coordinates.
02

Points of Intersection in Polar Graphs

The points of intersection of two polar graphs are the points (r, 胃) where the two graphs coincide with each other. Solving the equations simultaneously generally works, but might miss points in polar coordinates.
03

Further Analysis for Polar Intersection

To ensure all points of intersection are found, further analysis may be needed beyond merely solving the equations. This is due to the nature of polar coordinates, where the angle 胃 can be adjusted by adding or subtracting multiples of 2蟺 to give the same point. Therefore, you need to check for all possible values of r and 胃, not just those given by solving the equations.

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Most popular questions from this chapter

Finding Points of Intersection In Exercises \(25-32,\) find the points of intersection of the graphs of the equations. $$ \begin{array}{l}{r=1+\cos \theta} \\ {r=1-\sin \theta}\end{array} $$

Finding Points of Intersection In Exercises \(25-32,\) find the points of intersection of the graphs of the equations. $$ \begin{array}{l}{r=3+\sin \theta} \\ {r=2 \csc \theta}\end{array} $$

Conjecture Find the area of the region enclosed by $$r=a \cos (n \theta)$$ for \(n=1,2,3, \ldots\) Use the results to make a conjecture about the area enclosed by the function when \(n\) is even and when \(n\) is odd.

Sketching and Identifying a Conic In Exercises \(13-22\) , find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. $$ r=\frac{-6}{3+7 \sin \theta} $$

Conic What conic section does the polar equation \(r=a \sin \theta+b \cos \theta\) represent?
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