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Chapter 9: Problem 55

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

Short Answer

Expert verified
The polar equation \(r= a \sin \theta + b \cos \theta\) represents a straight line which is a degenerate conic section.

Step by step solution

01

Convert the polar equation to Cartesian

First, take note that in polar coordinates, \(x = r \cos \theta\) and \(y = r \sin \theta\). Now, replace \(r\), \(\sin \theta\) and \(\cos \theta\) with their Cartesian equivalents in the given equation.\[r = a \sin \theta + b \cos \theta 鈬� r = a * r * y + b * r * x\]This equation can be simplified to:\(r = r(ay + bx)\)Divide both sides of the equation by \(r (ay + bx)\) to express this equation in terms of Cartesian coordinates.\[(ay + bx) = 1\]
02

Identify the conic section

The Cartesian equation \(ay + bx = 1\) represents a straight line. In other words, the polar equation given in the problem is a straight line in polar coordinates. And a straight line is a conic section (degenerated).

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