Question

$A(3,2,0),B(5,3,2),C(-9,6,-3)$ are three points forming a triangle. If $AD$, the bisector of $\angle BAC$ meets $BC$ in $D$ then coordinates of $D$ are

• A. $(-\frac{19}{8},\frac{57}{16},\frac{17}{16})$
• B. $(\frac{19}{8},-\frac{57}{16},\frac{17}{16})$
• C. $(\frac{19}{8},\frac{57}{16},\frac{17}{16})$
• D. None of these

Answer 1

The correct option is A $(-\frac{19}{8},\frac{57}{16},\frac{17}{16})$

According to question,

$\begin{array}{c}AB=\sqrt{4+1+4}=3\\ AC=\sqrt{144+16+9}=13\\ BD:DC=AB:AC=3:13\\ D=\left(\frac{3(-9)+13\left(5\right)}{3+13},\frac{3\left(6\right)+13\left(3\right)}{3+13},\frac{3(-3)+13\left(2\right)}{3+13}\right)\\ =\left(\frac{-38}{16},\frac{57}{16},\frac{17}{16}\right)\\ \therefore \left(\frac{-19}{8},\frac{57}{16},\frac{17}{16}\right)\\ So,thecorrectoptionisA.\end{array}$