There are exactly two points on the ellipse `(x^2)/(a^2)+(y^2)/(b^2)`=1 whose distance from the centre of the ellipse are equal to `sqrt((3a^2-b^2)/(3))`. Eccentricity of this ellipse is

Options

A. `(1)/(2)`
B. `(sqrt1)/(2)`
C. `(1)/(sqrt3)`
D. `(1)/(sqrt2)`

Explanation

To solve the problem, we need to find the eccentricity of the ellipse given the condition about the distance from the center. Let's break it down step by step.

### Step 1: Understand the given ellipse equation
The equation of the ellipse is given by:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

### Step 2: Identify the distance from the center
We are given that the distance from the center of the ellipse to the points on the ellipse is:

\[ d = \sqrt{\frac{3a^2 - b^2}{3}} \]

### Step 3: Set the distance equal to the semi-minor axis
For points on the ellipse, the distance from the center to the ellipse along the y-axis (the semi-minor axis) is equal to \( b \). Therefore, we set:

\[ \sqrt{\frac{3a^2 - b^2}{3}} = b \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:

\[ \frac{3a^2 - b^2}{3} = b^2 \]

### Step 5: Cross-multiply to simplify
Cross-multiplying results in:

\[ 3a^2 - b^2 = 3b^2 \]

### Step 6: Rearrange the equation
Rearranging the equation gives:

\[ 3a^2 = 4b^2 \]

### Step 7: Express \( b^2 \) in terms of \( a^2 \)
From the equation \( 3a^2 = 4b^2 \), we can express \( \frac{b^2}{a^2} \):

\[ \frac{b^2}{a^2} = \frac{3}{4} \]

### Step 8: Use the formula for eccentricity
The eccentricity \( e \) of the ellipse is given by the formula:

\[ e^2 = 1 - \frac{b^2}{a^2} \]

Substituting \( \frac{b^2}{a^2} = \frac{3}{4} \):

\[ e^2 = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 9: Calculate the eccentricity
Taking the square root gives:

\[ e = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

### Conclusion
Thus, the eccentricity of the ellipse is:

\[ \boxed{\frac{1}{2}} \]