Question

If the pair of straight lines $x^2-2pxy-y^2=0$ and $x^2-2qxy-y^2=0$ be such that each pair bisects the angle between the other pair, then

• A. $p=-q$
• B. $pq=1$
• C. $pq=-1$
• D. $p=q$

Answer 1

The correct option is C $pq=-1$

Given equations are $x^2-2qxy-y^2=0...\left(1\right)$ $x^2-2pxy-y^2=0...\left(2\right)$ Joint equation of angle bisector of the line (i) and (ii) are same $\therefore q{x}^2+2xy-q{y}^2=\mathrm{0....}\left(3\right)$.

Now (2).and (3) are same, taking the ratio of their coefficients

$\therefore \frac{1}{q}=\frac{-p}{1}\Rightarrow pq=-1$