A point P is taken on the right half of the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` having its foci as `S_1 and S_2`. If the internal angle bisector of the angle `angleS_1PS_2` cuts the x-axis at poin `Q(alpha, 0)` then range of `alpha` is

Options

A. a. `[-a, a]`
B. b. `[0, a]`
C. c. `(0, a]`
D. d. `[-a, 0]`

Answer

To solve the problem step by step, we will analyze the given hyperbola and the conditions for the point \( Q(\alpha, 0) \).

### Step 1: Understand the Hyperbola
The equation of the hyperbola is given by:

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

This hyperbola opens to the right and has its foci at points \( S_1(-c, 0) \) and \( S_2(c, 0) \), where \( c = \sqrt{a^2 + b^2} \).

### Step 2: Identify the Point \( P \)
Point \( P \) lies on the right half of the hyperbola, meaning \( P \) has coordinates \( (x_P, y_P) \) where \( x_P > a \) (since it lies on the right side of the hyperbola).

### Step 3: Internal Angle Bisector
The internal angle bisector of angle \( S_1 P S_2 \) will divide the angle into two equal parts. The coordinates of point \( Q(\alpha, 0) \) where this bisector intersects the x-axis need to be determined.

### Step 4: Condition for Point \( Q \)
Since point \( Q \) lies on the x-axis, we need to consider the conditions under which \( Q \) is outside the hyperbola. For point \( Q(\alpha, 0) \) to be outside the hyperbola, it must satisfy:

\[ \frac{\alpha^2}{a^2} - \frac{0^2}{b^2} - 1 \leq 0 \]

This simplifies to:

\[ \frac{\alpha^2}{a^2} - 1 \leq 0 \]

### Step 5: Solve the Inequality
Rearranging the inequality gives:

\[ \frac{\alpha^2}{a^2} \leq 1 \]

Multiplying both sides by \( a^2 \) (assuming \( a > 0 \)) yields:

\[ \alpha^2 \leq a^2 \]

Taking the square root of both sides gives:

\[ -\alpha \leq a \quad \text{and} \quad \alpha \leq a \]

This leads to:

\[ -\sqrt{a^2} \leq \alpha \leq \sqrt{a^2} \]

Thus, we have:

\[ -\alpha \leq a \quad \text{and} \quad \alpha \leq a \]

### Step 6: Consider the Range of \( \alpha \)
Since \( Q \) is on the right half of the hyperbola, we also have the condition that \( \alpha > 0 \). Therefore, combining this with the previous result, we get:

\[ 0 < \alpha \leq a \]

### Final Result
The range of \( \alpha \) is:

\[ \alpha \in (0, a] \]