Question

A polar curve is represented by the equation r1 = 7 + 4cos θ.

Part A: What type of limaçon is this curve? Justify your answer using the constants in the equation.

Part B: Is the curve symmetrical to the polar axis or the line theta equals pi/2 Justify your answer algebraically.

Part C: What are the two main differences between the graphs of r1 = 7 + 4cos θ and r2 = 4 + 4cos θ?

Answer 1

a. dimpled: 4 < 7 < 2·4

b. not symmetrical about θ = π/2

c. r2: smaller, cardioid

Explanation:

The general form of the equation for a limaçon can be written ...

r = a·cos(θ) +b

This equation gives rise to 4 types of limaçon based on the relationship between a and b:

• looped: b < a
• cardioid: a = b
• dimpled: a < b < 2a
• convex: 2a ≤ b

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A.

We have a=4, b=7, so a < b < 2a. The limaçon is dimpled.

B.

The equation is not the same if θ is replaced by π/2 -θ. The curve is not symmetrical about θ = π/2.

C.

The equation for r2 ...

• gives a cardioid limaçon
• gives a limaçon with a smaller diameter (radial extent)