If the minor axis of an ellipse subtends an equilateral triangle with vertex  at one end of major axis, then write the eccentricity of the ellipse.

Options

A. `e=sqrt(2/3)`
B. `e=2/3`
C. `e=sqrt(1/3)`
D. `e=sqrt(3/4)`

Answer

To find the eccentricity of the ellipse given that the minor axis subtends an equilateral triangle with a vertex at one end of the major axis, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Equation of the Ellipse**:
The standard form of the ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \( a > b \). Here, \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

2. **Identify the Lengths of the Axes**:
The length of the major axis is \( 2a \) and the length of the minor axis is \( 2b \).

3. **Equilateral Triangle Properties**:
The problem states that the minor axis subtends an equilateral triangle. The side length of this triangle is equal to the length of the minor axis, which is \( 2b \).

4. **Height of the Equilateral Triangle**:
The height \( h \) of an equilateral triangle with side length \( s \) is given by:
\[ h = \frac{\sqrt{3}}{2} s \]
Substituting \( s = 2b \):
\[ h = \frac{\sqrt{3}}{2} \times 2b = \sqrt{3}b \]

5. **Relate the Height to the Semi-Major Axis**:
The height of the triangle also corresponds to the semi-major axis \( a \) of the ellipse. Therefore, we have:
\[ a = \sqrt{3}b \]

6. **Express \( a \) in terms of \( b \)**:
From the equation \( a = \sqrt{3}b \), we can express the ratio:
\[ \frac{a}{b} = \sqrt{3} \]
Squaring both sides gives:
\[ \frac{a^2}{b^2} = 3 \]

7. **Find \( \frac{b^2}{a^2} \)**:
Taking the reciprocal:
\[ \frac{b^2}{a^2} = \frac{1}{3} \]

8. **Calculate the Eccentricity**:
The eccentricity \( e \) of the ellipse is given by:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Substituting \( \frac{b^2}{a^2} = \frac{1}{3} \):
\[ e = \sqrt{1 - \frac{1}{3}} = \sqrt{\frac{2}{3}} \]

9. **Final Result**:
Therefore, the eccentricity of the ellipse is:
\[ e = \sqrt{\frac{2}{3}} \]