Consider the ellipse `x^(2)/(tan^(2)alpha)+y^(2)/(sec^(2)alpha)=1` where `alphain(0,pi/2)`. Which of the following quantities would vary as `alpha` varies?

Options

A. (a)degree of flatness
B. (b)ordinate of the vertex
C. (c)coordinate of the foci
D. (d)length of latusrectum

Answer

To solve the problem, we need to analyze the given equation of the ellipse and determine which quantities vary as the angle \(\alpha\) changes.

The equation of the ellipse is given as:

\[ \frac{x^2}{\tan^2 \alpha} + \frac{y^2}{\sec^2 \alpha} = 1 \]

### Step 1: Identify the semi-major and semi-minor axes

The standard form of the ellipse is:

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

From the given equation, we can identify:

- \(a^2 = \tan^2 \alpha\) ⇒ \(a = \tan \alpha\)
- \(b^2 = \sec^2 \alpha\) ⇒ \(b = \sec \alpha\)

### Step 2: Calculate the eccentricity \(e\)

The eccentricity \(e\) of an ellipse is given by the formula:

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

Substituting the values of \(a\) and \(b\):

\[ e = \sqrt{1 - \frac{\tan^2 \alpha}{\sec^2 \alpha}} = \sqrt{1 - \frac{\sin^2 \alpha}{\cos^2 \alpha}} = \sqrt{\frac{\cos^2 \alpha - \sin^2 \alpha}{\cos^2 \alpha}} = \sqrt{\frac{\cos^2 \alpha - (1 - \cos^2 \alpha)}{\cos^2 \alpha}} = \sqrt{\frac{2\cos^2 \alpha - 1}{\cos^2 \alpha}} \]

This shows that \(e\) depends on \(\alpha\).

### Step 3: Determine the coordinates of the foci

The coordinates of the foci of the ellipse are given by \((\pm ae, 0)\). Since \(a = \tan \alpha\) and \(e\) varies with \(\alpha\), the coordinates of the foci will also vary with \(\alpha\).

### Step 4: Calculate the length of the latus rectum

The length of the latus rectum \(L\) is given by:

\[ L = \frac{2b^2}{a} = \frac{2 \sec^2 \alpha}{\tan \alpha} \]

This quantity also varies with \(\alpha\).

### Conclusion

From the analysis, we find that the following quantities vary as \(\alpha\) varies:

1. The eccentricity \(e\)
2. The coordinates of the foci
3. The length of the latus rectum

### Summary of the Results

- **Eccentricity \(e\)** varies with \(\alpha\).
- **Coordinates of the foci** vary with \(\alpha\).
- **Length of the latus rectum** varies with \(\alpha\).