Question

If $Q$ be the foot of perpendicular from $P(2,4,3)$ on the line joining the points $A(1,2,4)$ and $B(3,4,5)$, then co-ordinates of $Q$ is given by

• A. $(\frac{19}{9},\frac{28}{9},\frac{41}{9})$
• B. $(\frac{28}{9},\frac{19}{9},\frac{41}{9})$
• C. $(\frac{41}{9},\frac{28}{9},\frac{19}{9})$
• D. $(0,0,0)$

Answer 1

The correct option is A $(\frac{19}{9},\frac{28}{9},\frac{41}{9})$

Equation of the line $AB$

$\frac{x-1}{3-1}=\frac{y-2}{4-2}=\frac{z-4}{5-4}$

$\frac{x-1}{2}=\frac{y-2}{2}=\frac{z-4}{1}$

Let,

$\frac{x-1}{2}=\frac{y-2}{2}=\frac{z-4}{1}=\lambda$

$x=2\lambda +1,y=2\lambda +2,z=\lambda +4$

General point on the line $AB$ is

$2\lambda +1,2\lambda +2,\lambda +4$

Let the coordinates of point $Q$ be $(2\lambda +1,2\lambda +2,\lambda +4)$

Direction ratios of the line $PQ$ are

$2\lambda +1-2,2\lambda +2-4,\lambda +4-3$

i.e.,

$2\lambda -1,2\lambda -2,\lambda +1$

Since line $PQ$ is perpendicular to line $AB$

So,

$2(2\lambda -1),2(2\lambda -2),1(\lambda +1)=0$

$\lambda =\frac{5}{9}$

Now,

$2\lambda +1=\frac{19}{9},2\lambda +2=\frac{28}{9},\lambda +4=\frac{41}{9}$

Therefore, coordinates of point is

$(\frac{19}{9},\frac{28}{9},\frac{41}{9})$

Hence, option $A$ is correct.