Maths > Straight Lines > 8.0 Equation of angle bisector
Straight Lines
1.0 Definition
2.0 Condition of collinearity of three points
3.0 Equation of a straight line in various forms
3.1 Point-slope form
3.2 Slope-Intercept form
3.3 Two point form
3.4 Determinant form
3.5 Intercept form
3.6 Normal form
3.7 Parametric form
4.0 Angle between two lines
5.0 Length of perpendicular from a point to a line
6.0 Foot of perpendicular from a point to a line
7.0 Reflection of a point about a line
7.1 Reflection with respect to $X-$axis
7.2 Reflection with respect to $Y-$axis
7.3 Reflection with respect to Origin
7.4 Reflection with respect to the line $x=a$
7.5 Reflection with respect to the line $y=b$
7.6 Reflection with respect to the line $y=x$
8.0 Equation of angle bisector
9.0 Bisector of angle containing origin
10.0 Bisector of angle containing a given point
11.0 Family of straight lines
8.1 Acute (internal) angle bisector and Obtuse (external) angle bisector
3.2 Slope-Intercept form
3.3 Two point form
3.4 Determinant form
3.5 Intercept form
3.6 Normal form
3.7 Parametric form
7.2 Reflection with respect to $Y-$axis
7.3 Reflection with respect to Origin
7.4 Reflection with respect to the line $x=a$
7.5 Reflection with respect to the line $y=b$
7.6 Reflection with respect to the line $y=x$
METHOD I:
- Find the equation of bisector of angle between the lines $ax + by + c = 0$ and $a'x + b'y + c' = 0$ using the formulae $$\frac{{{a}x + {b}y + {c}}}{{\sqrt {{a}^2 + {b}^2} }} = \pm \frac{{a'x + b'y + c'}}{{\sqrt {a{'^2} + b{'^2}} }}$$
- From here, we get two different equations of angle bisectors say, $(1)$ and $(2)$.
- Take equation $(1)$ and find the slope ${m_2}$.
- Take anyone equation of given line and finds its slope ${m_1}$.
- Find the angle $\theta $ between one of the given lines and one of the bisectors. Find $\tan \theta $.
- If $\left| {\tan \theta } \right| < 1,{\text{ }}\theta < {45^ \circ }$, this bisector is the acute angle bisector and othe bisector i.e., equation $(2)$ is the obtuse angle bisector.
- If $\left| {\tan \theta } \right| > 1,{\text{ }}\theta > {45^ \circ }$, this bisector is the obtuse angle bisector and other bisector i.e., equation $(2)$ is the acute angle bisector.
METHOD II: (Shortcut method)
- Let equation of two lines be $ax + by + c = 0$ and $a'x + b'y + c' = 0$. Taking ${c} > 0,{\text{ }}{c'} > 0{\text{ and }}{a}{b'} \ne {a'}{b}$.
- Find the equation of bisector of angle between the lines $ax + by + c = 0$ and $a'x + b'y + c' = 0$ using the formulae $$\frac{{{a}x + {b}y + {c}}}{{\sqrt {{a}^2 + {b}^2} }} = \pm \frac{{a'x + b'y + c'}}{{\sqrt {a{'^2} + b{'^2}} }}$$
| Conditions | Acute angle bisector | Obtuse angle bisector |
| $${a}{a'} + {b}{b'} > 0$$ | $-$ | $+$ |
| $${a}{a'} + {b}{b'} < 0$$ | $+$ | $-$ |
