Question

If a line drawn from the point $A(1,2,1)$ is perpendicular to the line joining $P(1,4,6)$ and $Q(5,4,4)$, then find the coordinates of the foot of the perpendicular

Answer 1

The equation of a line joining the points $P(1,4,6)$ and $Q(5,4,4)$ is
$\frac{x-1}{5-1}=\frac{y-4}{4-4}=\frac{z-6}{4-6}$
$\frac{x-1}{4}=\frac{y-4}{0}=\frac{z-6}{-2}$ .... (i)
Let $M$ be the foot of the perpendicular drawn from the point $A(1,2,1)$.
Coordinates of $M$ are given by,
$\frac{x-1}{4}=\frac{y-4}{0}=\frac{z-6}{-2}=\lambda$
$x=4\lambda +1,y=0\lambda +4,z=-2\lambda +6$
$\therefore M\equiv (4\lambda +1,4,-2\lambda +6)$ ..... (ii)
The direction ratios of $AM$ are $4\lambda +1-1,4-2,-2\lambda +6-1$
i.e., $4\lambda ,2,-2\lambda +5$
Direction ratios of given line are $(4,0,-2)$, since $AM$ is perpendicular to the given line.
$\therefore 4\left(4\lambda \right)+0\left(2\right)+(-2)(-2\lambda +5)=0$
$\therefore 16\lambda +4\lambda -10=0$
$\therefore \lambda =\frac{1}{2}$
Putting $\lambda =\frac{1}{2}$ in equation (ii), we get
$M\equiv (2,4,5)$
Hence, the co-ordinates of the foot of the perpendicular are $(3,4,5)$.