A hyperbola that opens left and right has a directrix located to the left of the pole (origin). In polar coordinates, the standard form for a hyperbola is given by:
\[ r = \frac{ed}{1 \mp e \cos \theta} \]
where:
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\( e \) is the eccentricity of the hyperbola, which is greater than 1,
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\( d \) is the distance from the pole (origin) to the directrix,
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The sign before \( e \cos \theta \) determines the direction of the opening:
- For left-opening hyperbolas: use the negative sign
- For right-opening hyperbolas: use the positive sign
Given that the vertical directrix is 1 unit to the left of the pole, we know that \( d = 1 \).
Now, also, since there is a directrix on the left side, the equation for the hyperbola should involve \(-\cos \theta\). Therefore, the form becomes:
\[ r = \frac{ed}{1 - e \cos \theta} \]
Comparing this setup with the options provided:
- \( r = \frac{3}{2 + 3 \sin \theta} \)
- \( r = \frac{3}{2 - 3 \sin \theta} \)
- \( r = \frac{3}{2 + 3 \cos \theta} \)
- \( r = \frac{3}{2 - 3 \cos \theta} \)
From the analysis, we can see that the equation for a hyperbola that opens left and right must follow the pattern where the numerator is related to \( d \) (1 unit) and the denominator has a negative sign in front of \( e \cos \theta \).
Given this context, we can choose the option that fits the equation of the hyperbola we derived.
The correct answer is:
\( r = \frac{3}{2 - 3 \cos \theta} \).