If the angle between two lines is `45^(@)` and the slope of one line is 2, then the product of possible slopes of other line is

Options

A. `-1`
B. 1
C. 2
D. 6

Answer

To find the product of the possible slopes of the other line when the angle between two lines is \(45^\circ\) and the slope of one line is \(m_1 = 2\), we can use the formula for the tangent of the angle between two lines.

### Step-by-step Solution:

1. **Understand the Given Information:**
- The angle between the two lines, \(\theta = 45^\circ\).
- The slope of one line, \(m_1 = 2\).

2. **Use the Tangent Formula:**
The formula for the tangent of the angle between two lines is given by:
\[ \tan(\theta) = \frac{m_1 - m_2}{1 + m_1 m_2} \]
For \(\theta = 45^\circ\), we have:
\[ \tan(45^\circ) = 1 \]
Thus, we can set up the equation:
\[ 1 = \frac{2 - m_2}{1 + 2m_2} \]

3. **Cross-Multiply to Solve for \(m_2\):**
Cross-multiplying gives:
\[ 1 + 2m_2 = 2 - m_2 \]

4. **Rearrange the Equation:**
Rearranging the equation:
\[ 2m_2 + m_2 = 2 - 1 \]
This simplifies to:
\[ 3m_2 = 1 \]
Therefore:
\[ m_2 = \frac{1}{3} \]

5. **Consider the Negative Case:**
The angle can also be negative, so we consider:
\[ -1 = \frac{2 - m_2}{1 + 2m_2} \]
Cross-multiplying gives:
\[ -1 - 2m_2 = 2 - m_2 \]

6. **Rearrange the Equation:**
Rearranging this equation:
\[ -2m_2 + m_2 = 2 + 1 \]
This simplifies to:
\[ -m_2 = 3 \]
Therefore:
\[ m_2 = -3 \]

7. **Find the Product of the Slopes:**
Now we have two possible values for \(m_2\):
\[ m_2 = \frac{1}{3} \quad \text{and} \quad m_2 = -3 \]
The product of the possible slopes is:
\[ \text{Product} = \left(\frac{1}{3}\right) \times (-3) = -1 \]

### Final Answer:
The product of the possible slopes of the other line is \(-1\).

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