Question

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

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 still aligns with lemniscate behavior).
• 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, \( 2n \) petals if \( n \) is even).

Step 2: Match Each Equation

• For \( r = a^{2}\sin(2\theta) \), the form (involving \( \sin(2\theta) \) and \( r \)) aligns with a lemniscate (since lemniscates often involve \( \sin(2\theta) \) or \( \cos(2\theta) \) with \( r^{2} \) or similar).
• For \( r = a \pm b\sin\theta \), this is the standard form of a limacon.
• For \( r = a\cos\theta \), this is the standard polar equation for a circle.
• For \( r = a\cos(n\theta) \), this is the standard form of a rose curve.

Answer:

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