If the sum of the slopes of the lines given by `x^(2)-2cxy-7y^(2)=0` is four times their product then c has the value

Options

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

Answer

To solve the problem, we need to find the value of \( c \) such that the sum of the slopes of the lines given by the equation \( x^2 - 2cxy - 7y^2 = 0 \) is four times their product.

### Step-by-Step Solution:

1. **Rewrite the Equation**:
The given equation is:
\[ x^2 - 2cxy - 7y^2 = 0 \]
This is a quadratic equation in terms of \( x \) and \( y \).

2. **Substitute \( y = mx \)**:
To find the slopes of the lines, we substitute \( y = mx \) into the equation:
\[ x^2 - 2cmx^2 - 7m^2x^2 = 0 \]
Factoring out \( x^2 \) (assuming \( x \neq 0 \)):
\[ x^2(1 - 2cm - 7m^2) = 0 \]
This gives us the equation:
\[ 1 - 2cm - 7m^2 = 0 \]

3. **Rearranging the Equation**:
Rearranging gives:
\[ 7m^2 + 2cm - 1 = 0 \]

4. **Identifying Coefficients**:
This is a standard quadratic equation in \( m \), where:
- \( a = 7 \)
- \( b = 2c \)
- \( c = -1 \)

5. **Using the Relationship Between Roots**:
According to the problem, the sum of the slopes \( m_1 + m_2 \) is equal to four times their product \( m_1 m_2 \). From the properties of quadratic equations:
- Sum of roots \( m_1 + m_2 = -\frac{b}{a} = -\frac{2c}{7} \)
- Product of roots \( m_1 m_2 = \frac{c}{a} = -\frac{1}{7} \)

6. **Setting Up the Equation**:
According to the problem statement:
\[ -\frac{2c}{7} = 4 \left(-\frac{1}{7}\right) \]
Simplifying the right side:
\[ -\frac{2c}{7} = -\frac{4}{7} \]

7. **Solving for \( c \)**:
Multiplying both sides by -7:
\[ 2c = 4 \]
Dividing both sides by 2:
\[ c = 2 \]

### Final Answer:
Thus, the value of \( c \) is:
\[ \boxed{2} \]