Question

The foot of the perpendicular from the origin to the join of $A(-9,4,5)$ and $B(11,0,-1)$ divides $AB$ in the ratio.

• A. $2:3$
• B. $3:2$
• C. $1:1$
• D. $1:2$

Answer 1

Answer: (C)

$1:1$

Let the point divide $AB$ in the ratio $k:1$.
The coordinates of the point will be $(\frac{11k-9}{k+1},\frac{4}{k+1},\frac{-k+5}{k+1})$
The dr's of the line joining the origin to this point are $(\frac{11k-9}{k+1},\frac{4}{k+1},\frac{-k+5}{k+1})$
The dr's of the line $AB$ are $(20,-4,-6)$.
Since, $AB$ has to be perpendicular to the line joining the other $2$ points,
$AB.OC=0$
$\Rightarrow 220k-180-16+6k-30=0$
$\Rightarrow k=1$
Hence, the line $AB$ is divided in the ratio $1:1$.