Straight Line and Pair of Straight Lines 2 Question 5
5. The vertices of a triangle are $A(-1,-7), B(5,1)$ and $C(1,4)$. The equation of the bisector of the angle $A B C$ is… .
$(1993,2 M)$
Analytical & Descriptive Questions
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Answer:
Correct Answer: 5. $7 y=x+2$
Solution:
- Equation of the line $A B$ is $y-1=\frac{1-(-7)}{5-(-1)}(x-5)$
$$ \begin{aligned} & \Rightarrow \quad y-1=\frac{8}{6}(x-5) \Rightarrow y-1=\frac{4}{3}(x-5) \\ & \Rightarrow \quad 3 y-3=4 x-20 \\ & \Rightarrow \quad 3 y-4 x+17=0 \end{aligned} $$
Equation of the line $B C$ is
$$ \begin{gathered} y-4=\frac{4-1}{1-5}(x-1) \quad \Rightarrow \quad y-4=-\frac{3}{4}(x-1) \\ \Rightarrow \quad 4 y-16=-3 x+3 \quad \Rightarrow \quad 3 x+4 y-19=0 \end{gathered} $$
Again, equation of the bisectors of the angles between two given lines $A B$ and $B C$ are
$$ \begin{aligned} & \frac{3 y-4 x+17}{\sqrt{3^{2}+4^{2}}}= \pm \frac{4 y+3 x-19}{\sqrt{4^{2}+3^{2}}} \\ & \Rightarrow \quad 3 y-4 x+17= \pm(4 y+3 x-19) \\ & \Rightarrow \quad 3 y-4 x+17=4 y+3 x-19 \\ & \text { and } \quad 3 y-4 x+17=-(4 y+3 x-19) \\ & \Rightarrow \quad 36=y+7 x \text { and } 7 y-x=2 \end{aligned} $$
Out of these two, equation of the bisector of angle $A B C$ is
$$ 7 y=x+2 \text {. } $$
