Question

The slope of one of the pair of lines \(2x^2 + hxy + 6y^2 = 0\) is three times the slope of the other line, \(h = ?\)

• A. \(16\)
• B. \(9\)
• C. \(18\)
• D. \(8\)

Hint:

For finding slopes of lines from a quadratic equation, use the standard forms for the pair of lines.
- Equating slopes helps in finding unknowns in line-related problems.

Answer 1

The Correct Option is D

Step 1: Find the slope of the lines.
For the given pair of lines, we know that the general equation is quadratic, and the slopes of the lines are related by the equation: \[ \text{Slope of first line} = m_1 = \frac{-h + \sqrt{h^2 - 24}}{4}, \quad \text{Slope of second line} = m_2 = \frac{-h - \sqrt{h^2 - 24}}{4}. \] Given that the slope of the first line is three times the slope of the second, we equate the slopes: \[ m_1 = 3 m_2 \quad \Rightarrow \quad \frac{-h + \sqrt{h^2 - 24}}{4} = 3 \times \frac{-h - \sqrt{h^2 - 24}}{4}. \] Step 2: Solve for \(h\).
Simplifying the equation, we get: \[ \boxed{h = 8}. \]