The eccentric angle of one end of a diameter of `x^(2)+3y^(2)=3` is `pi/6`, then the eccentric angle of the other end will be

Options

A. `(5pi)/6`
B. `-5pi/6`
C. `-2pi/3`
D. `2pi/3`

Answer

To solve the problem, we need to find the eccentric angle of the other end of the diameter of the ellipse given by the equation \(x^2 + 3y^2 = 3\), where one end of the diameter has an eccentric angle of \(\frac{\pi}{6}\).

### Step-by-Step Solution:

1. **Rewrite the Equation of the Ellipse**:
The given equation is:
\[ x^2 + 3y^2 = 3 \]
To convert it into standard form, divide the entire equation by 3:
\[ \frac{x^2}{3} + \frac{y^2}{1} = 1 \]
This shows that the semi-major axis \(a = \sqrt{3}\) and the semi-minor axis \(b = 1\).

**Hint**: To convert the equation of an ellipse into standard form, divide by the constant on the right side.

2. **Identify the Axes**:
From the standard form, we can see that the major axis is along the x-axis and the minor axis is along the y-axis.

**Hint**: Remember that the semi-major axis is the larger value when comparing \(a\) and \(b\).

3. **Understanding Eccentric Angles**:
The eccentric angle \(\theta\) is the angle formed with the positive x-axis. If one end of the diameter has an eccentric angle of \(\frac{\pi}{6}\), we need to find the angle of the other end of the diameter.

**Hint**: The eccentric angle is measured from the positive x-axis in the anticlockwise direction.

4. **Using the Linear Angle Property**:
The sum of angles on a straight line is \(\pi\) radians (or 180 degrees). If one end of the diameter makes an angle of \(\frac{\pi}{6}\) with the x-axis, the angle of the other end can be found using:
\[ \theta + \text{other angle} = \pi \]
Let the other angle be \(\theta'\):
\[ \frac{\pi}{6} + \theta' = \pi \]
Rearranging gives:
\[ \theta' = \pi - \frac{\pi}{6} \]

**Hint**: To find the other angle on a straight line, subtract the known angle from \(\pi\).

5. **Calculating the Other Angle**:
To find \(\theta'\):
\[ \theta' = \frac{6\pi}{6} - \frac{\pi}{6} = \frac{5\pi}{6} \]

**Hint**: Make sure to convert all terms to a common denominator when performing subtraction.

6. **Considering the Direction**:
Since the diameter is measured in the anticlockwise direction, we denote this angle as:
\[ -\frac{5\pi}{6} \]
This negative sign indicates the direction of measurement.

**Hint**: When measuring angles in a coordinate system, the direction (clockwise or anticlockwise) can affect the sign of the angle.

### Final Answer:
The eccentric angle of the other end of the diameter is:
\[ -\frac{5\pi}{6} \]