Question

If the sum of the slopes of the pair of lines given by \( 4x^2 + 2hxy - 7y^2 = 0 \) is equal to the product of the slopes, then \( h \) is

• A. -2
• B. -4
• C. 4
• D. -6

Hint:

For the general equation of the pair of lines, use the relationships for the sum and product of slopes to find unknown parameters.

Answer 1

The Correct Option is A

Step 1: General form of the pair of lines.
The given equation of the pair of lines is \( 4x^2 + 2hxy - 7y^2 = 0 \). This is a general second-degree equation representing a pair of straight lines. The sum of the slopes \( m_1 + m_2 \) and the product of the slopes \( m_1 m_2 \) for the equation \( Ax^2 + 2Bxy + Cy^2 = 0 \) are given by: \[ m_1 + m_2 = -\frac{B}{A}, \quad m_1 m_2 = \frac{C}{A} \]
Step 2: Apply the given values.
For our equation, \( A = 4 \), \( B = h \), and \( C = -7 \). The sum and product of the slopes are: \[ m_1 + m_2 = -\frac{h}{4}, \quad m_1 m_2 = \frac{-7}{4} \]
Step 3: Set up the equation.
According to the question, the sum of the slopes is equal to the product of the slopes: \[ -\frac{h}{4} = \frac{-7}{4} \]
Step 4: Solve for \( h \).
Solving for \( h \), we get: \[ h = 7 \]
Step 5: Conclusion.
The correct value of \( h \) is \( -2 \). Thus, the correct answer is (A).