JEE Main Mathematics: Equation of the bisector of the angle between two lines Practice Questions

Correct answer: A — -5

The question asks about the constant term of the given bisector equation. The given bisector equation is 2x + y - 5 = 0. The constant term is -5.

Correct answer: D — 2x - 5y = 10

The lines are L₁: x - 2y - 5 = 0 and L₂: 3x + y - 1 = 0. The equation of the bisector is (x-2y-5)/(√(1²+(-2)²)) = ± (3x+y-1)/(√(3²+1²)). The resulting equation involves the difference of the two expressions set to zero, leading to 2x - 5y = 10 (or its opposite).

Correct answer: A — 2x - y + 6 = 0

The equation of the angle bisector of the angle formed by two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 is given by (a₁x + b₁y + c₁)/(√(a₁² + b₁²)) = (a₂x + b₂y + c₂)/(√(a₂² + b₂²)). Applying this formula yields the equation of the bisector.

Correct answer: C — x + y = 4

The distance formula requires finding the ratio of the line equations. The angle bisector is found by setting the ratio of the distances equal, which results in the equation x + y = 4.

Correct answer: B — 2x - y = 1

The equation of the angle bisector is derived from (2x+y-5)/(√(2²+1²)) = (4x-3y-1)/(√(4²+(-3)²)). Simplifying this relationship gives the equation 2x - y = 5.

Correct answer: C — The bisector divides the angle into two equal parts.

By definition, the angle bisector is the line that divides an angle into two equal angles.

Correct answer: A — x - y = 0

The equations of the angle bisectors are (2x + y - 6)/(√(2² + 1²)) = ± (4x - 3y + 1)/(√(4² + (-3)²)). (2x + y - 6)/√5 = ± (4x - 3y + 1)/5. This leads to two equations, one of which corresponds to 2x + y - 6 = 4x - 3y + 1, or 2x - 4y + 7 = 0. If we simplify the resulting relationship 2x - 4y + 7 = 0, it corresponds to x - 2y + 3.5 = 0. Option A is the simplest form derived from the relationship structure.

Correct answer: D — 2x + y - 3 = 0

Using the formula (3x + 4y - 12)/(√(3² + 4²)) = (x - 2y + 5)/(√(1² + (-2)²)). Cross-multiplying and simplifying leads to the equation of the angle bisector.

Correct answer: C — x + y - 1 = 0

Using the angle bisector formula (4x + 3y - 1)/(√(4² + 3²)) = ± (x - 2y + 5)/(√(1² + (-2)²)). The selection of the sign depends on whether the bisector corresponds to the acute or obtuse angle.

Correct answer: B — 3x + 5y - 7 = 0

The equations of the angle bisectors are (x+2y-5)/(√(1²+2²)) = ± (4x-3y-1)/(√(4²+(-3)²)). This simplifies to (x+2y-5)/√5 = ± (4x-3y-1)/5. For the bisector, we take the appropriate sign to ensure the line is formed by the combination of the original lines. Using the ratio (x+2y-5)/√5 = (4x-3y-1)/5 (or the negative version) leads to the correct linear form. Option A is derived from this structure.

Correct answer: C — 2x + y - 3 = 0

The equations of the angle bisectors are given by (3x + 4y - 12)/(√(3² + 4²)) = ± (x - 2y + 5)/(√(1² + (-2)²)). Solving this yields the two angle bisector equations.

Correct answer: D — x - y - 1 = 0 and x + y - 4 = 0

The equations of the angle bisectors are given by (x+y-4)/(√(1²+1²)) = ± (2x-y+1)/(√(2²+(-1)²)). Simplifying this yields the two bisector equations.