Question

The direction ratios of the lines $AB$ and $AC$ are $(1,2,-2)$ and $(2,-3,6)$ respectively. The direction ratios of the angle bisector of the $\angle BAC$ are

• A. $3,-1,-8$
• B. $-1,5,-8$
• C. $1,23,-32$
• D. $1,23,32$

Answer 1

The correct option is C $1,23,-32$

Direction ratios of the $AB$ and $AC$ are $1,2,-2$ and $2,-3,6$ , then direction cosine are
For $AB$:
$l_1=\frac{1}{\sqrt{1+4+4}}=\frac{1}{3}$

$m_1=\frac{2}{\sqrt{1+4+4}}=\frac{2}{3}$

$n_1=\frac{-2}{\sqrt{1+4+4}}=-\frac{2}{3}$
For $AC$:
$l_2=\frac{1}{\sqrt{4+9+36}}=\frac{2}{7}$

$m_2=\frac{2}{\sqrt{4+9+36}}=-\frac{3}{7}$

$n_2=\frac{-2}{\sqrt{4+9+36}}=-\frac{6}{7}$
Now, direction cosines of the bisector of $BAC$ given by,
$l=l_1-l_2=\frac{1}{21},m=m_1-m_2=\frac{23}{21}$ and $n=n_1-n_2=\frac{-32}{21}$
Therefore, direction ratios are $(1,23,-32)$.
Hence, the option 'C' is correct.