To analyze the given polar equation

$r = \frac{4}{9 + 5 \cos \theta}$

we can determine the eccentricity, directrix, and shape of the curve by comparing this form to the standard polar equation of a conic section:

$r = \frac{ed}{1 + e \cos \theta}$

where:

• $e$ is the eccentricity,
• $d$ is the distance from the pole (origin) to the directrix.

Step 1: Identifying the Eccentricity

To match the given equation to the standard form, we rewrite it as:

$r = \frac{4}{9 + 5 \cos \theta} = \frac{\frac{4}{9}}{1 + \frac{5}{9} \cos \theta}$

From this form, we can see that:

• The coefficient of $\cos \theta$ in the denominator is $\frac{5}{9}$, which corresponds to the eccentricity $e$.

Thus, the eccentricity is:

$e = \frac{5}{9}$

Step 2: Identifying the Shape of the Curve

The shape of a conic section depends on the eccentricity $e$:

• If $0 \leq e < 1$, it is an ellipse.
• If $e = 1$, it is a parabola.
• If $e > 1$, it is a hyperbola.

Since $e = \frac{5}{9} < 1$, the curve is an ellipse.

Step 3: Finding the Directrix

In the polar form of a conic, $r = \frac{ed}{1 + e \cos \theta}$, the parameter $d$ is the distance to the directrix. From our form of the equation:

$\frac{ed}{1 + e \cos \theta} = \frac{4}{9}$

we identify that:

$ed = 4$

Solving for $d$:

$d = \frac{4}{e} = \frac{4}{\frac{5}{9}} = \frac{4 \times 9}{5} = \frac{36}{5}$

Thus, the equation of the directrix is:

$x = \frac{36}{5}$

Summary of Results

1. Eccentricity (e): $\frac{5}{9}$
2. Shape of the Curve: Ellipse
3. Equation of the Directrix: $x = \frac{36}{5}$

Would you like a detailed breakdown of any step? Here are some related questions:

1. How can eccentricity determine the shape of a conic section?
2. What happens if the eccentricity equals 1 in polar equations?
3. How is the directrix used to define conic sections?
4. What are the applications of polar equations in real-world scenarios?
5. Can we convert this polar equation to Cartesian coordinates?

Tip: Eccentricity values offer a quick way to identify conic shapes in polar and Cartesian forms.