If the sum of the slopes of the lines given by `x^(2)+2cxy-y^(2)=0` is four times their product, then c has the value

Options

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

Explanation

To solve the problem, we need to analyze the given equation of the pair of straight lines and use the relationship between the sum and product of the slopes.

### Step-by-Step Solution:

1. **Identify the given equation**:
The equation given is:
\[ x^2 + 2cxy - y^2 = 0 \]

2. **Recognize the form of the equation**:
This is a quadratic equation in terms of \(x\) and \(y\). It can be rewritten in the standard form of a pair of straight lines.

3. **Use the properties of the slopes**:
For a pair of straight lines represented by the equation \(Ax^2 + 2Bxy + Cy^2 = 0\), the sum of the slopes \(m_1 + m_2\) is given by:
\[ m_1 + m_2 = -\frac{B}{A} \]
and the product of the slopes \(m_1 m_2\) is given by:
\[ m_1 m_2 = \frac{C}{A} \]

4. **Identify coefficients**:
From the equation \(x^2 + 2cxy - y^2 = 0\):
- \(A = 1\)
- \(B = c\)
- \(C = -1\)

5. **Calculate the sum and product of the slopes**:
- The sum of the slopes:
\[ m_1 + m_2 = -\frac{B}{A} = -\frac{c}{1} = -c \]
- The product of the slopes:
\[ m_1 m_2 = \frac{C}{A} = \frac{-1}{1} = -1 \]

6. **Set up the equation based on the problem statement**:
According to the problem, the sum of the slopes is four times their product:
\[ m_1 + m_2 = 4(m_1 m_2) \]
Substituting the expressions we found:
\[ -c = 4(-1) \]
Simplifying this gives:
\[ -c = -4 \]

7. **Solve for \(c\)**:
\[ c = 4 \]

### Final Answer:
The value of \(c\) is \(4\).