The angular points of a triangle are $A(-1,-7)$ $B(5,1)$ and $C(1,4)$. The equation of the bisector of the angle $\angle A B C$ is

• A

  $x=7 y+2$
• Correct answer
  B

  $($ b) $7 y=x+2$
• C

  $y=7 x+2$
• D

  $7 x=y+2$

Solution

Here, $A B=\sqrt{(5+1)^{2}+(1+7)^{2}}=\sqrt{36+64}=10$ $B C=\sqrt{\left(1-5)^{2}+(4-1)^{2}\right.}=\sqrt{16+9}=5$ By angle bisector theorem, $$ A P: C P=10: 5=2: 1 $$ $P\left(\frac{2 \times 1+1 \times(-1)}{2+1}, \frac{2 \times 4+1 \times(-7)}{2+1}\right)=P\left(\frac{1}{3}, \frac{1}{3}\right)$ Required equation of $B P$ is $y-1=\frac{\frac{1}{3}-1}{\frac{1}{3}-5}(x-5)$ $\Rightarrow \quad y-1=\frac{-2}{-14}(x-5)$ $\Rightarrow \quad 7 y-7=x-5$ $\Rightarrow \quad 7 y=x+2$

Hence, the correct answer is option: 'B'.