Question

The equation of the line common to the pair of lines \[ (p^2 - q^2)x^2 + (q^2 - r^2)xy + (r^2 - p^2)y^2 = 0 \] and \[ (l - m)x^2 + (m - n)xy + (n - l)y^2 = 0 \] is:

• A. \( x + y = 0 \)
• B. \( x - y = 0 \)
• C. \( x + y = pqr \)
• D. \( x - y = pqr \)

Hint:

Look for symmetry in coefficients and test standard diagonals \( x = y \) or \( x = -y \) when identifying common lines.

Answer 1

The Correct Option is B

Each expression represents a pair of lines passing through the origin. The condition for a line to be common in both quadratic expressions is that it must satisfy both equations simultaneously. If you equate the general forms and compare their ratios, you'll find: \[ \frac{p^2 - q^2}{l - m} = \frac{q^2 - r^2}{m - n} = \frac{r^2 - p^2}{n - l} \Rightarrow \text{This implies symmetry about the line } x = y \Rightarrow \text{Common line is } x - y = 0 \]