What's the point?

Geometry Level 4

The line $\color{#D61F06}{l}$ has equations $\dfrac{x-1}{2}=\dfrac{y-1}{3}=\dfrac{z+1}{2}$ and the point $A$ is $(7, 3, 7)$ . $M$ is the point where the $\color{#007fff}{\text{perpendicular}}$ from $A$ meets $\color{#D61F06}{l}$ .

The point $\color{#ff8600}{B}$ lies on $AM$ such that $\overrightarrow { AB } =3 \, \overrightarrow { BM }$ . If the coordinates of $\color{#ff8600}{B}$ are $(p, q, r)$ , find the value of $pqr$ .

The answer is 132.

1 solution

Rishabh Jain
Feb 5, 2016

Let M be a general point on the line of the form (2r+1,3r+1,2r-1) (r∈ $\mathcal{R}$ ).
Now since AM is perpendicular to l , the dot product of their direction ratios must be 0 i.e
$(2r-6)2+(3r-2)3+(2r-8)2=0$ $\Rightarrow r=2$ Hence M≡ (5,7,3)
Now the point(B) which divides AM in the ratio 3:1 has coordinates p,q,r which are given by $p=\dfrac{3(5)+1(7)}{4}=\dfrac{11}{2}\\ q=\dfrac{3(7)+1(3)}{4}=6\\ r=\dfrac{3(3)+1(7)}{4}=4$ $\Large pqr=\dfrac{11}{2}\times 6\times 4=\boxed{132}$

What's the point?

The answer is 132.

1 solution

0 pending reports