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 can follow these steps:
### Step 1: Identify the equations
We have two equations:
1. The hyperbola: \(x^2 - y^2 = 1\)
2. The circle: \(x^2 + y^2 - 4x - 5 = 0\)
### Step 2: Rewrite the circle's equation
We can rewrite the circle's equation in standard form. First, rearranging gives:
\[
x^2 + y^2 - 4x - 5 = 0 \implies (x^2 - 4x) + y^2 = 5
\]
Completing the square for \(x\):
\[
(x^2 - 4x + 4) + y^2 = 5 + 4 \implies (x - 2)^2 + y^2 = 9
\]
This shows that the circle is centered at \((2, 0)\) with a radius of \(3\).
### Step 3: Find points of intersection
To find the points of intersection, we substitute \(y^2\) from the hyperbola's equation into the circle's equation. From the hyperbola, we have:
\[
y^2 = x^2 - 1
\]
Substituting this into the circle's equation:
\[
(x - 2)^2 + (x^2 - 1) = 9
\]
Expanding and simplifying:
\[
(x - 2)^2 + x^2 - 1 = 9
\]
\[
(x^2 - 4x + 4) + x^2 - 1 = 9
\]
\[
2x^2 - 4x + 3 - 9 = 0 \implies 2x^2 - 4x - 6 = 0
\]
Dividing the entire equation by 2:
\[
x^2 - 2x - 3 = 0
\]
### Step 4: Solve the quadratic equation
Now, we can solve the quadratic equation using the factorization method:
\[
(x - 3)(x + 1) = 0
\]
Thus, the solutions for \(x\) are:
\[
x = 3 \quad \text{and} \quad x = -1
\]
### Step 5: Find corresponding \(y\) values
Now, we find the corresponding \(y\) values for each \(x\):
1. For \(x = 3\):
\[
y^2 = 3^2 - 1 = 9 - 1 = 8 \implies y = \pm 2\sqrt{2}
\]
Thus, the points are \((3, 2\sqrt{2})\) and \((3, -2\sqrt{2})\).
2. For \(x = -1\):
\[
y^2 = (-1)^2 - 1 = 1 - 1 = 0 \implies y = 0
\]
Thus, the point is \((-1, 0)\).
### Step 6: Determine the number of distinct lines
Now we have three points of intersection:
1. \((3, 2\sqrt{2})\)
2. \((3, -2\sqrt{2})\)
3. \((-1, 0)\)
We can find the distinct lines through these points:
- The two points \((3, 2\sqrt{2})\) and \((3, -2\sqrt{2})\) give a vertical line.
- The point \((-1, 0)\) can connect to both \((3, 2\sqrt{2})\) and \((3, -2\sqrt{2})\), creating two additional lines.
Thus, we have a total of **3 distinct straight lines** through the points of intersection.
### Final Answer
The number of distinct straight lines through the points of intersection is **3**.
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