In an ellipse the distance between the foci is 8 and the distance between the directrices is 25. The length of major axis, is

Options

A. `10 sqrt2`
B. `20 sqrt2`
C. `30 sqrt2`
D. none of these

Answer

To solve the problem step by step, we will use the properties of an ellipse.

### Step 1: Understand the properties of the ellipse
In an ellipse, the distance between the foci (F1 and F2) is given by \(2ae\), where \(a\) is the semi-major axis and \(e\) is the eccentricity. The distance between the directrices is given by \(\frac{2a}{e}\).

### Step 2: Set up the equations
From the problem, we know:
1. The distance between the foci is 8:
\[ 2ae = 8 \quad \text{(Equation 1)} \]
This simplifies to:
\[ ae = 4 \]

2. The distance between the directrices is 25:
\[ \frac{2a}{e} = 25 \quad \text{(Equation 2)} \]
This simplifies to:
\[ \frac{a}{e} = 12.5 \]

### Step 3: Solve for \(a\) and \(e\)
From Equation 1, we can express \(e\) in terms of \(a\):
\[ e = \frac{4}{a} \]

Substituting this expression for \(e\) into Equation 2:
\[ \frac{a}{\frac{4}{a}} = 12.5 \]
This simplifies to:
\[ \frac{a^2}{4} = 12.5 \]
Multiplying both sides by 4:
\[ a^2 = 50 \]

### Step 4: Find \(a\)
Taking the square root of both sides:
\[ a = \sqrt{50} = 5\sqrt{2} \]

### Step 5: Calculate the length of the major axis
The length of the major axis is given by \(2a\):
\[ \text{Length of major axis} = 2a = 2 \times 5\sqrt{2} = 10\sqrt{2} \]

### Final Answer
Thus, the length of the major axis is \(10\sqrt{2}\).

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