Question

If the line \[ 2x + by + 5 = 0 \] forms an equilateral triangle with \[ ax^2 - 96bxy + ky^2 = 0, \] then \( a + 3k \) is: \

• A. \( 3b \)
• B. \( 192 \)
• C. \( 4b^2 \)
• D. \( 102 \)
  \

Hint:

For a line forming an equilateral triangle with two other lines, use the angle condition with the slopes and simplify using the given quadratic equation.

Answer 1

The Correct Option is B

Step 1: Understanding the Given Equation
The given quadratic equation: \[ ax^2 - 96bxy + ky^2 = 0 \] represents a pair of straight lines, which means it can be factored into two linear equations: \[ (ax + my)(bx + ny) = 0. \] The equation: \[ 2x + by + 5 = 0 \] forms an equilateral triangle with these lines. Step 2: Condition for an Equilateral Triangle
For three lines to form an equilateral triangle, the angle between the given line and one of the lines in the quadratic equation must be \( 60^\circ \). This means the slopes satisfy the angle condition: \[ \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| = \tan 60^\circ = \sqrt{3}. \] Using the quadratic equation in homogeneous form: \[ ax^2 - 96bxy + ky^2 = 0, \] the slopes of the pair of lines satisfy: \[ \text{Sum of slopes} = \frac{96b}{a+k}. \] For the given line, the slope is \( -\frac{2}{b} \). Step 3: Compute \( a + 3k \)
The equation simplifies using equilateral conditions, and after substituting values: \[ a + 3k = 192. \] Step 4: Conclusion
Thus, the final answer is: \[ \boxed{192}. \] \bigskip