To solve this problem, we need to determine the correct equation of the ellipse given the information about the distances between the foci and the directrices.

Step 1: Understand the relationship between parameters

• The general equation of an ellipse in standard form is:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

where:

• $2a$ is the length of the major axis.
• $2c$ is the distance between the foci, with $c = \sqrt{a^2 - b^2}$.
• The distance between the directrices is given by $\frac{2a^2}{\sqrt{a^2 - b^2}}$.

Step 2: Use the given distances

• The distance between the foci ($2c$) is given as 18, so $c = 9$.
• The distance between the directrices is given as 32, which means $\frac{2a^2}{c} = 32$.

Step 3: Find $a^2$ and $b^2$

1. From $c = 9$, we have $c^2 = 81$.

2. The relation $c^2 = a^2 - b^2$ gives us $a^2 - b^2 = 81$.

3. From $\frac{2a^2}{c} = 32$, we substitute $c = 9$: $\frac{2a^2}{9} = 32$ $2a^2 = 288$ $a^2 = 144$

4. Now, we know $a^2 - b^2 = 81$ and $a^2 = 144$. Therefore: $144 - b^2 = 81$ $b^2 = 63$

Step 4: Write the equation of the ellipse

The equation of the ellipse is: $\frac{x^2}{144} + \frac{y^2}{63} = 1$

Conclusion:

The correct answer is: Option 4) $\frac{x^2}{144} + \frac{y^2}{63} = 1$.

Would you like any more details or have any other questions?

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Here are 5 additional questions related to this problem:

1. How do you calculate the eccentricity of an ellipse given its major and minor axes?
2. What is the relationship between the lengths of the major axis, minor axis, and distance between foci in an ellipse?
3. How can the equation of a hyperbola be derived from the equation of an ellipse?
4. What are the properties of the directrices in an ellipse?
5. How does changing the length of the major axis affect the shape of an ellipse?

Tip: Remember that the sum of the distances from any point on an ellipse to the two foci is always constant and equal to the length of the major axis.