If the sum of the slopes of the lines given by `4x^2+2lambdaxy-7y^2=0` is equal to the product of the slope, then `lambda` is equal to

Options

A. a) -4
B. b) -2
C. c) 2
D. d) 4

Answer

To solve the given problem, we need to find the value of \( \lambda \) such that the sum of the slopes of the lines represented by the equation \( 4x^2 + 2\lambda xy - 7y^2 = 0 \) is equal to the product of the slopes.

### Step-by-Step Solution:

1. **Identify the Slopes**:
Let \( M_1 \) and \( M_2 \) be the slopes of the lines. According to the problem, we have:
\[ M_1 + M_2 = M_1 M_2 \]
We will denote this as Equation (1).

2. **Rewrite the Equation**:
The given equation is:
\[ 4x^2 + 2\lambda xy - 7y^2 = 0 \]
This is a quadratic equation in \( x \) and can be compared to the standard form of a conic section:
\[ Ax^2 + 2Hxy + By^2 = 0 \]
where \( A = 4 \), \( H = \lambda \), and \( B = -7 \).

3. **Use the Formulas for Slopes**:
The formulas for the sum and product of the slopes of the lines are:
\[ M_1 + M_2 = -\frac{2H}{B} \]
\[ M_1 M_2 = \frac{A}{B} \]

4. **Substitute Values**:
Substituting the values of \( A \), \( H \), and \( B \) into the formulas:
- For the sum:
\[ M_1 + M_2 = -\frac{2\lambda}{-7} = \frac{2\lambda}{7} \]
- For the product:
\[ M_1 M_2 = \frac{4}{-7} = -\frac{4}{7} \]

5. **Set Up the Equation**:
From Equation (1), we have:
\[ \frac{2\lambda}{7} = -\frac{4}{7} \]

6. **Clear the Denominator**:
Multiply both sides by 7 to eliminate the denominator:
\[ 2\lambda = -4 \]

7. **Solve for \( \lambda \)**:
Divide both sides by 2:
\[ \lambda = -2 \]

8. **Conclusion**:
The value of \( \lambda \) is \( -2 \).

### Final Answer:
Thus, the value of \( \lambda \) is:
\[ \lambda = -2 \]