Consider an ellipse, whose centre is at the origin and its major axis is along the x-axis. If its eccentricity is and the `3/5`distance between its foci is 6, then the area (in sq, units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is :

Options

A. 8
B. 32
C. 80
D. 40

Answer

To solve the problem step by step, we will use the properties of the ellipse and the given information.

### Step 1: Understand the properties of the ellipse
The equation of an ellipse centered at the origin with its major axis along the x-axis is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

### Step 2: Identify the given values
From the problem, we have:
- Eccentricity \( e = \frac{3}{5} \)
- Distance between the foci \( 2c = 6 \) which implies \( c = 3 \)

### Step 3: Relate eccentricity to the axes
The eccentricity \( e \) of an ellipse is given by:
\[ e = \frac{c}{a} \]
Substituting the known values:
\[ \frac{3}{5} = \frac{3}{a} \]
Cross-multiplying gives:
\[ 3a = 15 \implies a = 5 \]

### Step 4: Use the relationship between \( a \), \( b \), and \( c \)
We know the relationship:
\[ c^2 = a^2 - b^2 \]
Substituting the known values:
\[ 3^2 = 5^2 - b^2 \]
This simplifies to:
\[ 9 = 25 - b^2 \implies b^2 = 25 - 9 = 16 \implies b = 4 \]

### Step 5: Identify the vertices of the ellipse
The vertices of the ellipse are:
- Along the x-axis: \( (5, 0) \) and \( (-5, 0) \)
- Along the y-axis: \( (0, 4) \) and \( (0, -4) \)

### Step 6: Visualize the quadrilateral
The vertices of the quadrilateral inscribed in the ellipse are \( (5, 0) \), \( (0, 4) \), \( (-5, 0) \), and \( (0, -4) \). This forms a quadrilateral.

### Step 7: Calculate the area of the quadrilateral
The quadrilateral can be divided into four right triangles:
1. Triangle with vertices at \( (5, 0) \), \( (0, 4) \), and the origin \( (0, 0) \)
2. Triangle with vertices at \( (0, 4) \), \( (-5, 0) \), and the origin \( (0, 0) \)
3. Triangle with vertices at \( (-5, 0) \), \( (0, -4) \), and the origin \( (0, 0) \)
4. Triangle with vertices at \( (0, -4) \), \( (5, 0) \), and the origin \( (0, 0) \)

The area of one triangle can be calculated as:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the triangle with base \( 5 \) (from \( (5, 0) \) to \( (0, 0) \)) and height \( 4 \) (from \( (0, 0) \) to \( (0, 4) \)):
\[ \text{Area} = \frac{1}{2} \times 5 \times 4 = 10 \text{ square units} \]

### Step 8: Calculate the total area of the quadrilateral
Since there are four identical triangles:
\[ \text{Total Area} = 4 \times 10 = 40 \text{ square units} \]

### Final Answer
The area of the quadrilateral inscribed in the ellipse is \( 40 \) square units.

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