Straight Lines
10.0 Bisector of angle containing a given point
10.0 Bisector of angle containing a given point
Let equation of two lines be $ax + by + c = 0$ and $a'x + b'y + c' = 0$. Make ${c}$ and ${c'}$ positive.
The equation of bisector of angle between two lines containing the point $(h,k)$ will be $$\frac{{ax + by + c}}{{\sqrt {{a^2} + {b^2}} }} = \frac{{a'x + b'y + c'}}{{\sqrt {a{'^2} + b{'^2}} }}$$ or $$\frac{{ax + by + c}}{{\sqrt {{a^2} + {b^2}} }} = - \frac{{a'x + b'y + c'}}{{\sqrt {a{'^2} + b{'^2}} }}$$
according as $ah + bk + c$ and $a'h + b'k + c'$ are of same sign or opposite sign.
Question 6. For the straight lines $4x+3y-6=0$ and $5x+12y+9=0$, find the equation of bisector of obtuse angle and acute angle between them.
Solution: Write the equation of lines so that the constants become positive. $$\begin{equation} \begin{aligned} - 4x - 3y + 6 = 0...(1) \\ 5x + 12y + 9 = 0...(2) \\\end{aligned} \end{equation} $$
Here, ${a} = - 4,{\text{ }}{a'} = 5,{\text{ }}{b} = - 3,{\text{ }}{b'} = 12$
Now, ${a}{a'} + {b}{b'} = - 20 - 36 = - 56 < 0$
The equation of bisector of angle between the lines $ax + by + c = 0$ and $a'x + b'y + c' = 0$ using the formulae $$\frac{{{a}x + {b}y + {c}}}{{\sqrt {{a}^2 + {b}^2} }} = \pm \frac{{a'x + b'y + c'}}{{\sqrt {a{'^2} + b{'^2}} }}$$
Since, ${a}{a'} + {b}{b'}< 0$, the bisector of obtuse angle between the lines is given by $$\frac{{ax + by + c}}{{\sqrt {{a^2} + {b^2}} }} = - \frac{{a'x + b'y + c'}}{{\sqrt {a{'^2} + b{'^2}} }}$$$$\begin{equation} \begin{aligned} \frac{{ - 4x - 3y + 6}}{{\sqrt {{{( - 4)}^2} + {{( - 3)}^2}} }} = \frac{{5x + 12y + 9}}{{\sqrt {{5^2} + {{12}^2}} }} \\ 27x - 21y - 123 = 0 \\ 9x - 7y - 41 = 0 \\\end{aligned} \end{equation} $$
and the bisector of acute angle between the lines is given by $$\frac{{ax + by + c}}{{\sqrt {{a^2} + {b^2}} }} = \frac{{a'x + b'y + c'}}{{\sqrt {a{'^2} + b{'^2}} }}$$$$\begin{equation} \begin{aligned} \frac{{ - 4x - 3y + 6}}{{\sqrt {{{( - 4)}^2} + {{( - 3)}^2}} }} = \frac{{5x + 12y + 9}}{{\sqrt {{{(5)}^2} + {{(12)}^2}} }} \\ 77x + 99y - 33 = 0 \\ 7x + 9y - 3 = 0 \\\end{aligned} \end{equation} $$
