Question

The co-ordinates of the foot of perpendicular drawn from the origin to the line joining the points (-9, 4, 5) and (10, 0, -1) will be

• A. (- 3, 2, 1)
• B. (1, 2, 2)
• C. (4, 5, 3)
• D. None of these

Answer 1

The correct option is D

None of these

Let AD be perpendicular and D be foot of perpendicular which divide
BC in ratio $\lambda :1$ then

The direction ration of AD are $\frac{10\lambda -9}{\lambda +1},\frac{4}{\lambda +1},\frac{-\lambda +5}{\lambda +1}$ and direction ratio of BC are 19, -4 and -6.
Since AD $\perp$ BC
$\Rightarrow 19\left(\frac{10\lambda -9}{\lambda +1}\right)-4\left(\frac{4}{\lambda +1}\right)-6\left(\frac{-\lambda +5}{\lambda +1}\right)=0$
$\Rightarrow \lambda =\frac{31}{28}$
Hence on putting the value of $\lambda$ in (i), we get required foot of the perpendicular i.e., $(\frac{58}{59},\frac{112}{59},\frac{109}{59})$.
Trick : The line passing through these points is $\frac{x+9}{19}=\frac{y-4}{-4}=\frac{z-5}{-6}$. Now co-ordinates of the foot lie on this line, so they must satisfy the given line. But here no point satisfies the line, hence answer is (d).