If the normals at `alpha, beta,gamma` and `delta` on an ellipse are concurrent then the value of `(sigma cos alpha)(sigma sec alpha)` I

Options

A. 2
B. 4
C. 6
D. none of these

Explanation

To solve the problem, we need to find the value of \((\sigma \cos \alpha)(\sigma \sec \alpha)\) given that the normals at angles \(\alpha, \beta, \gamma,\) and \(\delta\) on an ellipse are concurrent.

### Step-by-Step Solution:

1. **Equation of the Ellipse**:
The standard equation of the ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

**Hint**: Remember that \(a\) and \(b\) are the semi-major and semi-minor axes of the ellipse, respectively.

2. **Equation of the Normal**:
The equation of the normal at any point \(P(\theta)\) on the ellipse can be expressed as:
\[ a \sec \theta (x - a \sec \theta) + b \csc \theta (y - b \csc \theta) = 0 \]

**Hint**: The normal line is derived from the slope of the tangent at the point and uses trigonometric identities.

3. **Concurrent Normals**:
If the normals at angles \(\alpha, \beta, \gamma,\) and \(\delta\) are concurrent, it means that there exists a common point through which all these normals pass.

4. **Using Properties of Normals**:
For the normals to be concurrent, we can use the condition involving the summation of the secants and cosines of the angles:
\[ \sigma \sec \theta = \sum \sec \theta \]
and
\[ \sigma \cos \theta = \sum \cos \theta \]

5. **Finding the Value**:
The value we are looking for is:
\[ (\sigma \cos \alpha)(\sigma \sec \alpha) \]
Substituting the expressions we derived:
\[ = \left(\sum \cos \theta\right) \left(\sum \sec \theta\right) \]

6. **Calculating the Result**:
From the properties of the ellipse and the conditions of concurrency, we can derive that:
\[ \sigma \cos \alpha \cdot \sigma \sec \alpha = 4 \]

**Hint**: This result comes from the relationships established through the geometry of the ellipse and the properties of the angles involved.

### Final Answer:
Thus, the value of \((\sigma \cos \alpha)(\sigma \sec \alpha)\) is:
\[ \boxed{4} \]