Question

The pairs of straight lines \( 2x^2 + axy + 3y^2 = 0 \) and \( 2x^2 + bxy - 3y^2 = 0 \) share a common line, and the other lines are perpendicular. Then values of \( a \) and \( b \) are?

• A. \( -5, 1 \)
• B. \( 5, -1 \)
• C. \( 5, 1 \)
• D. \( 5, \frac{1}{5} \)

Hint:

For homogeneous second degree equations representing pairs of lines, use discriminants and compare slopes to establish perpendicularity or parallelism.

Answer 1

The Correct Option is C

Let line 1: \( 2x^2 + axy + 3y^2 = 0 \) Discriminant \( D_1 = a^2 - 4(2)(3) = a^2 - 24 \) Similarly, line 2: \( 2x^2 + bxy - 3y^2 = 0 \), discriminant \( D_2 = b^2 + 24 \) The condition: they share one line, so one linear factor is common. If one pair has slope \( m_1 \), and the other lines are perpendicular: \[ m_1 m_2 = -1 \Rightarrow m_1 \text{ from first, } m_2 \text{ from second (non-shared)} \] Using condition from symmetric quadratic forms: Shared line means one linear factor is common. So compare slopes: Find condition under which one line is common and the other pair is perpendicular. This yields: \[ a = 5,\quad b = 1 \]