Question

The pairs of straight lines $a{x}^2+2hxy-a{y}^2=0$ and $h{x}^2-2axy-h{y}^2=0$ are such that

• A. one pair bisects the angles between the other pair
• B. the lines of one pair are equally inclined to the lines of the other pair
• C. the lines of one pair are perpendicular to the `lines of the other pair
• D. none of these

Answer 1

Answer: (A), (B)

The correct options are
A one pair bisects the angles between the other pair
B the lines of one pair are equally inclined to the lines of the other pair

Given pairs of straight lines are $a{x}^2+2hxy-a{y}^2=0$ and $h{x}^2-2axy-h{y}^2=0$.

Pair of angular bisectors of $a{x}^2+2hxy-a{y}^2=0$ is $h(x^2-y^2)=(a-(-a\left)\right)xy$

$\Rightarrow h{x}^2-2axy-h{y}^2$

$\therefore$ One pair bisects the angle between the other.

Clearly, if one pair bisects the angle between the other, the lines of one pair are equally inclined to the lines of the other pair.
Hence, option A and B.