The number of distinct straight lines through the points of intersection of `x^(2) - y^(2) = 1" and " x^(2) + y^(2) - 4x - 5 = 0 `

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the number of distinct straight lines through the points of intersection of the hyperbola \( x^2 - y^2 = 1 \) and the circle \( x^2 + y^2 - 4x - 5 = 0 \), we will follow these steps:

### Step 1: Find the points of intersection
We need to solve the equations of the hyperbola and the circle simultaneously.

1. The equation of the hyperbola is:
\[ x^2 - y^2 = 1 \tag{1} \]

2. The equation of the circle can be rearranged. First, we complete the square:
\[ x^2 + y^2 - 4x - 5 = 0 \implies (x-2)^2 + y^2 = 9 \]
This represents a circle with center \( (2, 0) \) and radius \( 3 \).

### Step 2: Substitute \( y^2 \) from the hyperbola into the circle's equation
From equation (1), we can express \( y^2 \):
\[ y^2 = x^2 - 1 \]
Now, substitute \( y^2 \) into the circle's equation:
\[ (x-2)^2 + (x^2 - 1) = 9 \]

### Step 3: Simplify and solve for \( x \)
Expanding the equation:
\[ (x-2)^2 + x^2 - 1 = 9 \]
\[ (x^2 - 4x + 4) + x^2 - 1 = 9 \]
Combine like terms:
\[ 2x^2 - 4x + 3 = 9 \]
\[ 2x^2 - 4x - 6 = 0 \]
Divide the entire equation by 2:
\[ x^2 - 2x - 3 = 0 \]

### Step 4: Factor the quadratic equation
Factoring gives:
\[ (x - 3)(x + 1) = 0 \]
Thus, the solutions for \( x \) are:
\[ x = 3 \quad \text{and} \quad x = -1 \]

### Step 5: Find corresponding \( y \) values
Substituting \( x = 3 \) into \( y^2 = x^2 - 1 \):
\[ y^2 = 3^2 - 1 = 9 - 1 = 8 \implies y = \pm 2\sqrt{2} \]
So we have the points \( (3, 2\sqrt{2}) \) and \( (3, -2\sqrt{2}) \).

Now substituting \( x = -1 \):
\[ y^2 = (-1)^2 - 1 = 1 - 1 = 0 \implies y = 0 \]
So we have the point \( (-1, 0) \).

### Step 6: Determine the number of distinct lines
The points of intersection are:
1. \( (3, 2\sqrt{2}) \)
2. \( (3, -2\sqrt{2}) \)
3. \( (-1, 0) \)

To find the number of distinct straight lines through these points:
- From \( (3, 2\sqrt{2}) \) and \( (3, -2\sqrt{2}) \), we can draw a vertical line.
- From \( (3, 2\sqrt{2}) \) and \( (-1, 0) \), we can draw a line.
- From \( (3, -2\sqrt{2}) \) and \( (-1, 0) \), we can draw another line.

Thus, we have a total of **three distinct lines**.

### Final Answer
The number of distinct straight lines through the points of intersection is **3**.

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