Problem

Given the polar equation \( r = 1 - 2\cos(\theta) \), identify the type of conic section and sketch the graph.

Solution

Step 1 :Step 1: Observe the given equation. In polar coordinates, the equation of a cardioid can be written as \( r = a(1 \pm \cos(\theta)) \) or \( r = a(1 \pm \sin(\theta)) \).

Step 2 :Step 2: Comparing the given equation with the standard form, it is clear that the given equation represents a cardioid. The value of 'a' is -2, which means the cardioid opens in the negative x-direction.

Step 3 :Step 3: The graph of a cardioid opening in the negative x-direction starts at the pole (0,0) and extends left along the x-axis. The point farthest from the pole is 2 units to the left of the pole.

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Source: https://solvelyapp.com/problems/wc9zlyGOHC/

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