Question

The equation \((2p - 3)x^2 + 2pxy - y^2 = 0\) represents a pair of distinct lines

• A. Only when \(p = 0\)
• B. \(p \in \mathbb{R} - \{-3,1\}\)
• C. For all values of \(p \in \mathbb{R} - [-3,1]\)
• D. For all values of \(p \in \mathbb{R}\)

Hint:

Use discriminant condition \(h^2 - ab>0\) to identify pair of distinct lines.

Answer 1

The Correct Option is B

Pair of straight lines: Discriminant \(> 0\) \Rightarrow check determinant \(\Delta = abc + 2fgh - af^2 - bg^2 - ch^2\)
Or use \(D = h^2 - ab>0\) where \(a = 2p - 3, b = -1, h = p\)
\[ D = p^2 + (2p - 3)>0 \Rightarrow p^2 + 2p - 3>0 \Rightarrow (p + 3)(p - 1)>0 \Rightarrow p \in (-\infty, -3) \cup (1, \infty) \Rightarrow \text{Exclude } -3, 1 \]