Problem

Given the polar equation r=1−2cos⁡(θ), identify the type of conic section and sketch the graph.

Answer

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Answer

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.

Steps

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

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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