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 distance from the center to certain points on the ellipse. Here’s a step-by-step solution:

### Step 1: Understand the given ellipse equation
The equation of the ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \( a \) is the semi-major axis and \( b \) is the semi-minor axis. We assume \( a > b \).

### Step 2: Set up the distance condition
The problem states that there are exactly two points on the ellipse whose distance from the center (0,0) is equal to:
\[ d = \sqrt{\frac{3a^2 - b^2}{3}} \]
We know that the distance from the center to any point on the ellipse can be expressed as:
\[ d = \sqrt{x^2 + y^2} \]

### Step 3: Relate the distance to the ellipse
Since the points are on the ellipse, we can express the distance in terms of \( b \) (the semi-minor axis):
\[ d = b \]
Thus, we have:
\[ \sqrt{\frac{3a^2 - b^2}{3}} = b \]

### Step 4: Square both sides
To eliminate the square root, we square both sides:
\[ \frac{3a^2 - b^2}{3} = b^2 \]
Multiplying both sides by 3 gives:
\[ 3a^2 - b^2 = 3b^2 \]
Rearranging this, we find:
\[ 3a^2 = 4b^2 \]

### Step 5: Solve for \( b^2 \)
From the equation \( 3a^2 = 4b^2 \), we can express \( b^2 \) in terms of \( a^2 \):
\[ b^2 = \frac{3a^2}{4} \]

### Step 6: Use the relationship between \( a^2 \), \( b^2 \), and eccentricity
We know that the relationship between \( a^2 \), \( b^2 \), and the eccentricity \( e \) of the ellipse is given by:
\[ b^2 = a^2(1 - e^2) \]
Substituting \( b^2 = \frac{3a^2}{4} \) into this equation, we have:
\[ \frac{3a^2}{4} = a^2(1 - e^2) \]

### Step 7: Simplify the equation
Dividing both sides by \( a^2 \) (assuming \( a \neq 0 \)):
\[ \frac{3}{4} = 1 - e^2 \]
Rearranging gives:
\[ e^2 = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 8: Find the eccentricity
Taking the square root of both sides, we find:
\[ e = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

### Conclusion
Thus, the eccentricity of the ellipse is:
\[ \boxed{\frac{1}{2}} \]