Question

The equation of the line through the point $(0,1,2)$ and perpendicular to the line
$\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2}$ is :

• A. $\frac{x}{-3}=\frac{y-1}{4}=\frac{z-2}{3}$
• B. $\frac{x}{3}=\frac{y-1}{4}=\frac{z-2}{3}$
• C. $\frac{x}{3}=\frac{y-1}{-4}=\frac{z-2}{3}$
• D. $\frac{x}{3}=\frac{y-1}{4}=\frac{z-2}{-3}$

Answer 1

The correct option is A $\frac{x}{-3}=\frac{y-1}{4}=\frac{z-2}{3}$

Let $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2}=\lambda$
Any point $B$ on this line is $(2\lambda +1,3\lambda -1,-2\lambda +1)$

direction ratio of given line ${\overrightarrow{d}}_1\equiv (2,3,-2)$
Direction ratio of line to be found ${\overrightarrow{d}}_2\equiv \overrightarrow{AB}\equiv (2\lambda +1,3\lambda -2,-2\lambda -1)$
$\therefore {\overrightarrow{d}}_1.{\overrightarrow{d}}_2=0$
$\Rightarrow 4\lambda +2+9\lambda -6+4\lambda +2=0$
$\Rightarrow \lambda =\frac{2}{17}$
Direction ratio of line $(21,-28,-21)\equiv (3,-4,-3)\equiv (-3,4,3)$
As line passes through $(0,1,2)$
So, equation of line is
$\frac{x}{-3}=\frac{y-1}{4}=\frac{z-2}{3}$