Question

The co-ordinates of the foot of perpendicular drawn from point P(1,0,3) to the line joining the points A(4,7,1) and B(3,5,3) is

• A. $(5,7,1)$
• B. $(\frac{5}{3},\frac{7}{3},\frac{17}{3})$
• C. $(\frac{2}{3},\frac{5}{3},\frac{7}{3})$
• D. $(\frac{5}{3},\frac{2}{3},\frac{7}{3})$

Answer 1

The correct option is B $(\frac{5}{3},\frac{7}{3},\frac{17}{3})$

Let D be the foot of perpendicular drawn from P(1,0,3) on the line AB joining (4, 7, 1) and (3, 5, 3). If D divides AB in ratio $\lambda :1$ then $D=(\frac{3\lambda +4}{\lambda +1},\frac{5\lambda +7}{\lambda +1},\frac{3\lambda +1}{\lambda +1}).....\left(i\right)$

D.r's of PD are $2\lambda +3,5\lambda +7,-2$
D.r's of AB are -1, -2,2
$\because PD\perp AB;\therefore -(2\lambda +3)-2(5\lambda +7)-4=0\Rightarrow \lambda =\frac{-7}{4}$
Putting the value of $\lambda$ in (i), we get the point $D(\frac{5}{3},\frac{7}{3},\frac{17}{3})$.