Area of the quadrilateral formed by the extremities of major axis and minor axis is `8sqrt(3)`. The distance between foci is `4sqrt(2)`. Then eccentricity of the ellipse is

Options

A. `(1)/sqrt(3)`
B. `1/3`
C. `sqrt(2/3)`
D. `2/3`

Answer

To find the eccentricity of the ellipse given the area of the quadrilateral formed by the extremities of the major and minor axes and the distance between the foci, we can follow these steps:

### Step 1: Understand the properties of the ellipse
The area of the quadrilateral formed by the extremities of the major axis (2a) and minor axis (2b) of the ellipse is given by the formula:
\[ \text{Area} = 2a \cdot 2b = 4ab \]
Given that the area is \(8\sqrt{3}\), we can set up the equation:
\[ 4ab = 8\sqrt{3} \]

### Step 2: Solve for \(ab\)
Dividing both sides of the equation by 4:
\[ ab = 2\sqrt{3} \]

### Step 3: Use the distance between the foci
The distance between the foci of the ellipse is given by the formula:
\[ 2c \]
where \(c = \sqrt{a^2 - b^2}\). We are given that this distance is \(4\sqrt{2}\), so:
\[ 2c = 4\sqrt{2} \implies c = 2\sqrt{2} \]

### Step 4: Relate \(c\), \(a\), and \(b\)
From the relationship \(c = \sqrt{a^2 - b^2}\), we can square both sides:
\[ c^2 = a^2 - b^2 \implies (2\sqrt{2})^2 = a^2 - b^2 \implies 8 = a^2 - b^2 \]

### Step 5: Set up the equations
Now we have two equations:
1. \(ab = 2\sqrt{3}\)
2. \(a^2 - b^2 = 8\)

### Step 6: Express \(a\) in terms of \(b\)
From the first equation, we can express \(a\) as:
\[ a = \frac{2\sqrt{3}}{b} \]

### Step 7: Substitute \(a\) in the second equation
Substituting \(a\) in the second equation:
\[ \left(\frac{2\sqrt{3}}{b}\right)^2 - b^2 = 8 \]
This simplifies to:
\[ \frac{12}{b^2} - b^2 = 8 \]

### Step 8: Multiply through by \(b^2\)
To eliminate the fraction, multiply through by \(b^2\):
\[ 12 - b^4 = 8b^2 \]
Rearranging gives:
\[ b^4 + 8b^2 - 12 = 0 \]

### Step 9: Let \(x = b^2\)
Let \(x = b^2\), then the equation becomes:
\[ x^2 + 8x - 12 = 0 \]

### Step 10: Solve the quadratic equation
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot (-12)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ = \frac{-8 \pm \sqrt{64 + 48}}{2} = \frac{-8 \pm \sqrt{112}}{2} = \frac{-8 \pm 4\sqrt{7}}{2} = -4 \pm 2\sqrt{7} \]

### Step 11: Find positive \(b^2\)
We take the positive root:
\[ b^2 = -4 + 2\sqrt{7} \]

### Step 12: Find \(a^2\)
Now, substituting back to find \(a^2\):
\[ a^2 = b^2 + 8 = (-4 + 2\sqrt{7}) + 8 = 4 + 2\sqrt{7} \]

### Step 13: Calculate eccentricity
The eccentricity \(e\) is given by:
\[ e = \frac{c}{a} = \frac{\sqrt{a^2 - b^2}}{a} \]
Calculating \(e\):
\[ e = \frac{2\sqrt{2}}{\sqrt{4 + 2\sqrt{7}}} \]

### Final Step: Simplify
To simplify, we can calculate \(e\) numerically or leave it in this form.