Identifying the Acute-Angle Bisector

To determine which of the two bisectors (from x²−y² = 0 or xy = 0 form) makes the acute angle with the original lines: check the sign of h(a−b). The result avoids solving the full bisector equations.

The bisector equation $(x^2-y^2)/(a-b) = xy/h$ represents two lines. To identify which is the acute-angle bisector:

The two bisectors emerge from:

$$h(x^2-y^2) = (a-b)xy$$

This is one combined equation. The two individual bisectors are obtained by solving alongside $ax^2+2hxy+by^2=0$.

Rule: The acute-angle bisector is:

• The bisector containing the $xy$-term's contribution when $h(a-b) < 0$
• The bisector from the $x^2-y^2$ contribution when $h(a-b) > 0$

More practical test: For the two lines $L_1$ and $L_2$, the bisector of the acute angle between them is the one that makes an angle less than 45° with either line.

Origin test (for bisectors of lines not through origin): For the general equation, after finding both bisector equations, substitute the origin $(0,0)$ into both. The origin lies in the obtuse angle if $a+b > 0$ and $\Delta < 0$ (specific sign analysis). In JEE problems, it is typically faster to substitute a test point.

Practical approach: Compute $h(a-b)$:

• If $h(a-b) > 0$: the bisector $h(x^2-y^2) - (a-b)xy = 0$ that makes a smaller angle with the x-axis is the obtuse bisector.
• If $h(a-b) < 0$: it is the acute bisector.

When in doubt, substitute a point on one of the original lines into both bisector equations and determine which bisector is closer.