Question

If x² - 2pxy - y² = 0 and x² - 2qxy - y² = 0 bisect angles between each other, then

1. p + q = 1
2. pq = 1
3. pq + 1 = 0
4. p² + pq + q² = 0

Answer 1

Correct Answer - Option 3 : pq + 1 = 0

Concept:

The equation for the angle bisector for ax² + 2hxy + by² = 0 is given by

\(\frac{x^{2}-y^{2}}{a-b}=\frac{xy}{h}\)

Given:

x2 - 2pxy - y2 = 0 and x2 - 2qxy - y2 = 0 bisect angles between each other.

Calculation:

The equation of the bisectors of the angles between the lines x² - 2pxy - y² = 0

\(\dfrac{x^2 - y^2}{1-(-1)} = \dfrac{xy}{-p}\)

\(\Rightarrow \dfrac{x^2 - y^2}{2} = \dfrac{-xy}{p}\)

\(\Rightarrow px^2 + 2xy - py^2 = 0\)

This is same as  x2 - 2qxy - y2 = 0. Therefore

\(\dfrac{p}{1} = \dfrac{2}{-2q} = \dfrac{p}{1}\)

⇒ pq + 1 = 0