Axes are co-ordinate axes, A and B are ends of major axes and minor axis, area of `triangle`OAB is 16 sq units if e=`sqrt(3)/(2)` then equation of the ellipse is

Options

A. `(x^(2))/(32)+y^2/8=1`
B. `(x^(2))/(64)+(y^(2))/(16)=1`
C. `(x^(2))/(64)+(y^(2))/(8)=1`
D. `(x^(2))/(64)+(y^(2))/(32)=1`

Answer

To solve the problem step by step, we will derive the equation of the ellipse given the area of triangle OAB and the eccentricity.

### Step 1: Understand the Geometry
The points A and B are the ends of the major and minor axes of the ellipse, respectively. The origin O is the center of the ellipse. The area of triangle OAB can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is \(2A\) (the length of the major axis) and the height is \(2B\) (the length of the minor axis).

### Step 2: Set Up the Area Equation
Given that the area of triangle OAB is 16 square units, we can write:
\[ \frac{1}{2} \times 2A \times 2B = 16 \]
This simplifies to:
\[ 2AB = 16 \implies AB = 8 \]

### Step 3: Use the Eccentricity
The eccentricity \(e\) is given as \(\frac{\sqrt{3}}{2}\). The relationship between the semi-major axis \(A\), semi-minor axis \(B\), and eccentricity \(e\) is:
\[ e = \sqrt{1 - \frac{B^2}{A^2}} \]
Substituting the value of \(e\):
\[ \frac{\sqrt{3}}{2} = \sqrt{1 - \frac{B^2}{A^2}} \]

### Step 4: Square Both Sides
Squaring both sides gives:
\[ \left(\frac{\sqrt{3}}{2}\right)^2 = 1 - \frac{B^2}{A^2} \]
This simplifies to:
\[ \frac{3}{4} = 1 - \frac{B^2}{A^2} \]
Rearranging gives:
\[ \frac{B^2}{A^2} = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 5: Relate B to A
From \(\frac{B^2}{A^2} = \frac{1}{4}\), we can express \(B\) in terms of \(A\):
\[ B = \frac{A}{2} \]

### Step 6: Substitute into the Area Equation
Now substitute \(B = \frac{A}{2}\) into the equation \(AB = 8\):
\[ A \left(\frac{A}{2}\right) = 8 \]
This simplifies to:
\[ \frac{A^2}{2} = 8 \implies A^2 = 16 \implies A = 4 \]

### Step 7: Find B
Using \(B = \frac{A}{2}\):
\[ B = \frac{4}{2} = 2 \]

### Step 8: Write the Equation of the Ellipse
The standard form of the ellipse is:
\[ \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1 \]
Substituting \(A = 4\) and \(B = 2\):
\[ \frac{x^2}{16} + \frac{y^2}{4} = 1 \]

### Final Answer
The equation of the ellipse is:
\[ \frac{x^2}{16} + \frac{y^2}{4} = 1 \]