First slide

Equation of line in Straight lines

Question

Two lines are intersecting at the point $(1, 1)$ . If $2 x + 3 y - 5 = 0$ is the equation of one angular bisector of the two lines and the equation of the other bisector is $lx + my + n = 0$ then $l + m + n$ is

Moderate

A

2
B

3
C

1
D

0

Solution

The angular bisector of any pair of lines are passing through the point of intersection of those two lines and the angular bisectors are mutually perpendicular to each other.
The other angular bisector is a line passing through $(1, 1)$ and perpendicular to the line $2 x + 3 y - 5 = 0$
The equation of any line passing through $(x_{1}, y_{1})$ and perpendicular to the line $ax + by + c = 0$ is $b (x - x_{1}) - a (y - y_{1}) = 0$
Therefore, the equation of the other angular bisector is $3 (x - 1) - 2 (y - 1) = 0$
Rewrite the equation as $3 x - 2 y - 1 = 0$
Therefore, $l + m + n = 3 - 2 - 1 = 0$

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