The distance between the directrices of the ellipse `(x^(2))/(36)+(y^(2))/(20)=1` is

Options

A. 16 units
B. 17 units
C. 18 units
D. 19 units

Answer

To find the distance between the directrices of the ellipse given by the equation \(\frac{x^2}{36} + \frac{y^2}{20} = 1\), we will follow these steps:

### Step 1: Identify \(a^2\) and \(b^2\)
From the equation of the ellipse, we can identify:
- \(a^2 = 36\)
- \(b^2 = 20\)

### Step 2: Calculate \(a\) and \(b\)
We can find \(a\) and \(b\) by taking the square root of \(a^2\) and \(b^2\):
- \(a = \sqrt{36} = 6\)
- \(b = \sqrt{20} = 2\sqrt{5}\)

### Step 3: Calculate the eccentricity \(e\)
The eccentricity \(e\) of an ellipse is given by the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Substituting the values of \(a^2\) and \(b^2\):
\[ e = \sqrt{1 - \frac{20}{36}} = \sqrt{1 - \frac{5}{9}} = \sqrt{\frac{4}{9}} = \frac{2}{3} \]

### Step 4: Find the distance between the directrices
The equations of the directrices of an ellipse are given by:
\[ x = \pm \frac{a}{e} \]
Thus, the distance between the directrices is:
\[ \text{Distance} = 2 \cdot \frac{a}{e} \]
Substituting the values of \(a\) and \(e\):
\[ \text{Distance} = 2 \cdot \frac{6}{\frac{2}{3}} = 2 \cdot 6 \cdot \frac{3}{2} = 18 \]

### Final Answer
The distance between the directrices of the ellipse is \(18\).

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