Question

If the points $A(1,0,-4),B(-1,p,q)$ and $C(-3,7,4)$ are collinear,find the values of $p$ and $q$

Answer 1

Given that the points $A(1,0,-4),B(-1,p,q)$ and $C(-3,7,4)$ are collinear.Let the point $B$ divide the line segment $AC$ in the ratio $k:1$, then by section formula the coordinates of the point $B$ are

$(\frac{-3k+1}{k+1},\frac{7k+0}{k+1},\frac{4k-4}{k+1})$

But the coordinates of the point $B$ are $(-1,p,q)$, so we have,

$\frac{-3k+1}{k+1}=-1,\frac{7k}{k+1}=p,\frac{4k-4}{k+1}=q$

$\Rightarrow -3k+1=-k-1,\frac{7k}{k+1}=p,\frac{4k-4}{k+1}=q$

$\Rightarrow -3k+k=-1-1,\frac{7k}{k+1}=p,\frac{4k-4}{k+1}=q$

$\Rightarrow -2k=-2,\frac{7k}{k+1}=p,\frac{4k-4}{k+1}=q$

$\Rightarrow k=1,\frac{7\times 1}{1+1}=p,\frac{4\times 1-4}{1+1}=q$

$\Rightarrow k=1,\frac{7}{2}=p,\frac{0}{2}=q$

$\therefore k=1,p=\frac{7}{2},q=0$