Question

In $\Delta ABC$ if $A=(-1,2,-3),B=(5,0,-6)$ and $C=(O,4,-1)$, then the the directions of the internal bisector of $\angle BAC$ is

• A. $\frac{25}{10},\frac{8}{10},\frac{5}{10}$
• B. $\frac{-25}{10},\frac{8}{10},\frac{5}{10}$
• C. $\frac{25}{10},\frac{-8}{10},\frac{5}{10}$
• D. $\frac{25}{10},\frac{8}{10},\frac{-5}{10}$

Answer 1

The correct option is A $\frac{25}{10},\frac{8}{10},\frac{5}{10}$

Consider the lines $BC$ and $AB$ in vector from
$BC=5i-4j-5k$
$AB=6i-2j-3k$
Let the vector equation $\left(\lambda \right)$ of the angle bisector will be $ai+bj+ck$
Taking dot product, we get
$AB.\left(\lambda \right)$ implies
$6a-2b-3c=7\sqrt{a^2+b^2+c^2}cos\theta$ ...(i)
$BC.\left(\right(\lambda )$ implies
$5a-4b-5c=\sqrt{66}\sqrt{a^2+b^2+c^2}cos\theta$ ...(ii)
By comparing (ii) and (i), we get
$\frac{6a-2b-3c}{7}=\frac{5a-4b-5c}{\sqrt{66}}$
On simplifying, we get
$(6\sqrt{66}-35)a+(28-2\sqrt{66})b+(35-3\sqrt{66})c=0$
Since $a,b$ and $c$ are direction ratios, by matching the values of $a,b,c$ from the above options (which satisfies the above equation), we can get the values of $a,b,c$.