Maths · JEE Advanced only

Equation of Angle Bisector Between Two Lines (JEE Advanced)

Not in the JEE Main syllabus, but it is in JEE Advanced. Study time: about 3 to 4 hours for concept and formula mastery, plus 2 to 3 hours of practice problems. Total around 5 to 7 hours spread over a few days..

Last updated: 26 October 2026

Direct answer

The equation of angle bisector between two lines gives you the locus of all points equidistant from both lines.

The equation of angle bisector between two lines gives you the locus of all points equidistant from both lines. If two lines intersect, they form four angles, and there are exactly two angle bisectors: one for the acute angle and one for the obtuse angle. These bisectors are always perpendicular to each other.

Official syllabus

What the JEE Advanced syllabus says

“equation of the bisector of the angle between two lines”

Source: JEE Advanced 2026 Information Bulletin, Annexure-I. Closest JEE Main topic: angles between two lines, conditions for concurrence of three lines.

JEE Main only tests angles between lines and concurrence conditions, but Advanced expects you to find and use the actual bisector equations. You need this for locus problems, reflection questions, and combined geometry problems that appear regularly in Advanced.

Key ideas

What you must understand

The angle bisector is the set of all points whose perpendicular distances from both lines are equal
Two intersecting lines always have two bisectors, and these bisectors are perpendicular to each other
The bisector containing the origin can be identified by checking signs of constants in the line equations
To find which bisector is acute or obtuse, compare the product of slopes or use the angle condition
Angle bisectors are useful in finding incenters and excenters of triangles
Reflection of a point about a line lies on the line joining the point to its image, which involves bisector concepts

Formulas

Formulas and key facts

For lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, bisectors are: (a₁x + b₁y + c₁)/√(a₁² + b₁²) = ±(a₂x + b₂y + c₂)/√(a₂² + b₂²)
If c₁ and c₂ have the same sign, the + sign gives the bisector of the angle containing the origin
If c₁ and c₂ have opposite signs, the − sign gives the bisector of the angle containing the origin
For acute angle bisector: if a₁a₂ + b₁b₂ > 0, use the − sign; if a₁a₂ + b₁b₂ < 0, use the + sign
The two angle bisectors are always perpendicular (their slopes multiply to −1)
Incenter lies on internal angle bisectors; excenter lies on one internal and two external bisectors

Avoid

Common mistakes

Forgetting to make c₁ and c₂ both positive (or both negative) before applying the origin-containing bisector rule
Confusing acute angle bisector with the bisector containing the origin, they are not the same thing
Not simplifying the bisector equation after cross-multiplying, leading to messy calculations
Using the formula without checking if lines are parallel, parallel lines have no angle bisector through intersection

Worked example

A JEE Advanced-level example

Question. Find the equation of the bisector of the acute angle between the lines 3x − 4y + 7 = 0 and 12x + 5y − 2 = 0.

Show solution

Write the bisector formula: (3x − 4y + 7)/√(9 + 16) = ±(12x + 5y − 2)/√(144 + 25)
Simplify denominators: (3x − 4y + 7)/5 = ±(12x + 5y − 2)/13
Check for acute bisector: a₁a₂ + b₁b₂ = (3)(12) + (−4)(5) = 36 − 20 = 16 > 0, so use the − sign
Cross multiply with − sign: 13(3x − 4y + 7) = −5(12x + 5y − 2)
Expand: 39x − 52y + 91 = −60x − 25y + 10
Simplify: 99x − 27y + 81 = 0, which gives 11x − 3y + 9 = 0

Answer: 11x − 3y + 9 = 0

Solved twice independently; both solutions agree.

FAQs

Frequently asked questions

Is Equation of Angle Bisector Between Two Lines in the JEE Main syllabus?

No. It is in the JEE Advanced 2026 syllabus only. The closest JEE Main topic is angles between two lines, conditions for concurrence of three lines.

How long does Equation of Angle Bisector Between Two Lines take to prepare for JEE Advanced?

About 3 to 4 hours for concept and formula mastery, plus 2 to 3 hours of practice problems. Total around 5 to 7 hours spread over a few days., if you already know the related JEE Main topic.