Identify whether the given equation is that of a cardioid, limaçon, rose, or lemniscate.
$r=3+2\cos\theta$

• A. Cardioid
• B. Limacon
• C. Rose
• D. Lemniscate

Answer: (B)

Verified step by step guidance

1. Step 1: Recognize the general form of polar equations. A limaçon typically has the form r = a + b cos(θ) or r = a + b sin(θ).
2. Step 2: Compare the given equation r = 3 + 2 cos(θ) with the general form of a limaçon. Here, a = 3 and b = 2.
3. Step 3: Understand the characteristics of a limaçon. If |a| > |b|, the limaçon is without an inner loop. If |a| = |b|, it is a cardioid. If |a| < |b|, it has an inner loop.
4. Step 4: Calculate |a| and |b| for the given equation. Here, |a| = 3 and |b| = 2, so |a| > |b|, indicating a limaçon without an inner loop.
5. Step 5: Conclude that the given equation r = 3 + 2 cos(θ) represents a limaçon without an inner loop based on the comparison and characteristics.