For the hyperbola `x^2/ cos^2 alpha - y^2 /sin^2 alpha = 1;(0 lt alphalt pi/4)`. Which of the following remains constant when alpha varies?

Options

A. Eccentricity
B. Abscissa of foci
C. Directrix
D. Vertex

Explanation

To solve the problem, we need to analyze the hyperbola given by the equation:

\[ \frac{x^2}{\cos^2 \alpha} - \frac{y^2}{\sin^2 \alpha} = 1 \]

where \(0 < \alpha < \frac{\pi}{4}\). We will determine which of the parameters (centricity, abscissa of foci, directrix, and vertex) remains constant as \(\alpha\) varies.

### Step 1: Identify the parameters of the hyperbola
The standard form of a hyperbola is given by:

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

From the given equation, we can identify:
- \(a = \cos \alpha\)
- \(b = \sin \alpha\)

### Step 2: Calculate the eccentricity (e)
The eccentricity \(e\) of a hyperbola is given by:

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

Substituting \(a\) and \(b\):

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

### Step 3: Analyze the eccentricity
Since \(\sec \alpha\) varies as \(\alpha\) changes, the eccentricity is not constant.

### Step 4: Find the coordinates of the foci
The foci of the hyperbola are located at:

\[ (\pm ae, 0) = (\pm \cos \alpha \cdot \sec \alpha, 0) = (\pm 1, 0) \]

### Step 5: Analyze the foci
The coordinates of the foci are \((\pm 1, 0)\), which are independent of \(\alpha\). Thus, the abscissa of the foci remains constant.

### Step 6: Find the vertices
The vertices of the hyperbola are located at:

\[ (\pm a, 0) = (\pm \cos \alpha, 0) \]

### Step 7: Analyze the vertices
The vertices depend on \(\alpha\) since \(\cos \alpha\) varies with \(\alpha\).

### Step 8: Find the directrix
The directrix of the hyperbola is given by:

\[ x = \pm \frac{a}{e} = \pm \frac{\cos \alpha}{\sec \alpha} = \pm \cos^2 \alpha \]

### Step 9: Analyze the directrix
The directrix also depends on \(\alpha\) since \(\cos^2 \alpha\) varies with \(\alpha\).

### Conclusion
After analyzing all the parameters:
- Centricity varies with \(\alpha\)
- Abscissa of foci is constant (equal to 1)
- Directrix varies with \(\alpha\)
- Vertex varies with \(\alpha\)

Thus, the parameter that remains constant when \(\alpha\) varies is:

**Abscissa of Foci**