Question

match each polar equation with its correct shape. r = a² sin(2θ) choose... r = a ± b sinθ choose... r = a cosθ lemniscate limacon circle rose r = a cos(nθ)

Explanation:

Step 1: Recall Polar Curve Definitions

• Lemniscate: Polar equations of the form \( r^{2}=a^{2}\sin(2\theta) \) (or \( r^{2}=a^{2}\cos(2\theta) \)) represent lemniscates. The given \( r = a^{2}\sin(2\theta) \) can be rewritten as \( r^{2}=a^{4}\sin(2\theta) \) (though typically \( r^{2}=a^{2}\sin(2\theta) \) is standard, the structure here aligns with lemniscate form).
• Limacon: Polar equations of the form \( r = a \pm b\sin\theta \) (or \( r = a \pm b\cos\theta \)) represent limaçons (with special cases like cardioids when \( a = b \)).
• Circle: The equation \( r = a\cos\theta \) (or \( r = a\sin\theta \)) represents a circle with diameter \( a \) (centered at \( (\frac{a}{2}, 0) \) for \( r = a\cos\theta \) in polar coordinates).
• Rose: Polar equations of the form \( r = a\cos(n\theta) \) (or \( r = a\sin(n\theta) \)) represent rose curves, with \( n \) petals if \( n \) is odd, or \( 2n \) petals if \( n \) is even.

Step 2: Match Each Equation to Its Shape

1. \( r = a^{2}\sin(2\theta) \): Matches the lemniscate form (due to the \( \sin(2\theta) \) term and squared-like structure).
2. \( r = a \pm b\sin\theta \): Matches the limacon form (standard limacon equation).
3. \( r = a\cos\theta \): Matches the circle form (standard polar circle equation).
4. \( r = a\cos(n\theta) \): Matches the rose curve form (standard rose curve equation).

Answer:

1. \( r = a^{2}\sin(2\theta) \): lemniscate
2. \( r = a \pm b\sin\theta \): limacon
3. \( r = a\cos\theta \): circle
4. \( r = a\cos(n\theta) \): rose