Question

The coordinates of the foot of the 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

Answer: (D)

Let $AD$ be the perpendicular and $D$ be the foot of the perpendicular which divides $BC$ in the ratio $\lambda :1$, then
$D\left(\frac{10\lambda -9}{\lambda +1},\frac{4}{\lambda +1},\frac{-\lambda +5}{\lambda +1}\right)\dots (i)$

The direction ratios of $AD$ are $\frac{10\lambda -9}{\lambda +1},\frac{4}{\lambda +1}$ and $\frac{-\lambda +5}{\lambda +1}$
and direction ratios of BC are 1 9 ,-4 and - 6
Since $AD\perp BC$, we get
$19\left(\frac{10\lambda -9}{\lambda +1}\right)-4\left(\frac{4}{\lambda +1}\right)-6\left(\frac{-\lambda +5}{\lambda +1}\right)=0$
or $\lambda =\frac{31}{28}$
Hence, on putting the value of $\lambda$ in (i), we get required foot of the perpendicular, i.e., $\left(\frac{58}{59},\frac{112}{59},\frac{109}{59}\right)$