Question

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

Short answer

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

Step-by-step solution

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.\[r = a \sin \theta + b \cos \theta ⇒ r = a * r * y + b * r * x\]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.\[(ay + bx) = 1\]

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).