Straight Lines
Angle Bisectors of Pair of Lines
AIEEE 2003
Grade 11
Question:
If the pair of straight lines $x^2 - 2pxy - y^2 = 0$ and $x^2 - 2qxy - y^2 = 0$ are such that each pair bisects the angles between the other pair, then $pq$ equals:
(1) $1$
(2) $-1$
(3) $2$
(4) $-2$
Step-by-Step Solution
Key Concept: For a homogeneous pair $ax^2+2hxy+by^2 = 0$, the angle bisectors satisfy $(x^2-y^2)/(a-b) = xy/h$. For the first pair, find the bisector pair and compare it with the second pair.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)