Question

In a triangle $ABC$, right angled at the vertex $A$, if the position vectors of $A,B$ and $C$ are respectively $3\hat{i}+\hat{j}-\hat{k},-\hat{i}+3\hat{j}+p\hat{k}$ and $5\hat{i}+q\hat{j}-4\hat{k}$, then the point $(p,q)$ lies on a line :

• A. parallel to $x$-axis
• B. parallel to $y$-axis.
• C. making an acute angle with the positive direction of $x$ -axis.
• D. making an obtuse angle with the positive direction of $x$-axis.

Answer 1

Answer: (C)

$\overline{AB}=-4\hat{i}+2\hat{j}+\left(p+1\right)\hat{k}$
$\overline{AB}=2\hat{i}+\left(a_n-1\right)\hat{J}-3\hat{k}$
$\overline{AB}\perp \overline{AC}$
$\Rightarrow \overline{AB}.\overline{AC}=0$
$-8+2\left(a_n-1\right)-3\left(p+1\right)=0$
$3p-2q+13=0$
$\left(p,q\right)$lies on $3x-2y+13=0$
slope$=\frac{3}{2}$
$\therefore$ Acute angle with x - axis