Question

If the points $A(4,2,2),B(4,p,q)$ and $C(2,3,3)$ are collinear,find the values of $p$ and $q$

Answer 1

Given that the points $A(4,2,2),B(4,p,q)$ and $C(2,3,3)$ 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

$(\frac{2k+4}{k+1},\frac{3k+2}{k+1},\frac{3k+2}{k+1})$

But the coordinates of the point $B$ are $(4,p,q)$, so we have,

$\frac{2k+4}{k+1}=4,\frac{3k+2}{k+1}=p,\frac{3k+2}{k+1}=q$

$\Rightarrow 2k+4=4k+4,\frac{3k+2}{k+1}=p,\frac{3k+2}{k+1}=q$

$\Rightarrow 2k-4k=4-4,\frac{3k+2}{k+1}=p,\frac{3k+2}{k+1}=q$

$\Rightarrow -2k=0,\frac{3k+2}{k+1}=p,\frac{3k+2}{k+1}=q$

$\Rightarrow k=0,\frac{3\times 0+2}{0+1}=p,\frac{3\times 0+2}{0+1}=q$

$\therefore k=0,p=2,q=2$