To solve the problem of finding the locus of the intersection of two straight lines passing through the points (1, 0) and (-1, 0) respectively, and including an angle of 45 degrees, we can follow these steps:
### Step 1: Define the points and slopes
Let the intersection point of the two lines be (H, K). The slopes of the lines passing through the points (1, 0) and (-1, 0) can be defined as follows:
- For the line through (1, 0):
\[
M_1 = \frac{K - 0}{H - 1} = \frac{K}{H - 1}
\]
- For the line through (-1, 0):
\[
M_2 = \frac{K - 0}{H + 1} = \frac{K}{H + 1}
\]
### Step 2: Use the angle between the lines
Since the angle between the two lines is 45 degrees, we can use the formula for the tangent of the angle between two lines:
\[
\tan(\theta) = \left| \frac{M_1 - M_2}{1 + M_1 M_2} \right|
\]
For \(\theta = 45^\circ\), \(\tan(45^\circ) = 1\). Therefore, we have:
\[
1 = \left| \frac{\frac{K}{H - 1} - \frac{K}{H + 1}}{1 + \frac{K}{H - 1} \cdot \frac{K}{H + 1}} \right|
\]
### Step 3: Simplify the equation
Now, simplifying the left-hand side:
\[
\frac{K}{H - 1} - \frac{K}{H + 1} = K \left( \frac{(H + 1) - (H - 1)}{(H - 1)(H + 1)} \right) = K \left( \frac{2}{H^2 - 1} \right)
\]
And for the denominator:
\[
1 + \frac{K^2}{(H - 1)(H + 1)} = \frac{(H^2 - 1) + K^2}{H^2 - 1}
\]
Thus, we can rewrite the equation as:
\[
1 = \left| \frac{K \cdot 2}{H^2 - 1 + K^2} \right|
\]
### Step 4: Set up the equations
Cross-multiplying gives us:
\[
H^2 - 1 + K^2 = 2K \quad \text{or} \quad H^2 - 1 + K^2 = -2K
\]
### Step 5: Rearranging both cases
1. **First case**:
\[
H^2 + K^2 - 2K - 1 = 0 \implies H^2 + (K - 1)^2 = 2
\]
This represents a circle with center (0, 1) and radius \(\sqrt{2}\).
2. **Second case**:
\[
H^2 + K^2 + 2K - 1 = 0 \implies H^2 + (K + 1)^2 = 2
\]
This represents a circle with center (0, -1) and radius \(\sqrt{2}\).
### Conclusion
The locus of the intersection points of the two lines is a circle centered at either (0, 1) with radius \(\sqrt{2}\) or (0, -1) with radius \(\sqrt{2}\).
### Final Answer
Thus, the correct options are:
- (c) center (0, 1) and radius \(\sqrt{2}\)
- (d) center (0, -1) and radius \(\sqrt{2}\)
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