Question

The point $(0,-1,-1),(4,5,1),(3,9,4)$ and $(-4,4,k)$are coplaner, then k =

Answer 1

Given points are $(0,-1,-1),(4,5,1),(3,9,4)$

$and(-4,4,k)are$ coplanar.

$LetA=(0,-1,-1),B=(4,5,1),$

$C=(3,9,4),D=(-4,4,k)$

$\overrightarrow{AB}=(4-0,5+1,1+1)=(4,6,2)$

$\overrightarrow{AC}=(3-0,9+1,4+1)=(3,10,5)$

$\overrightarrow{AD}=(-4-0,4+1,k+1)=(-4,5,k+1)$

$\left|\begin{array}{ccc}4& 6& 2\\ 3& 10& 5\\ -4& 5& k+1\end{array}\right|=0$

$⟹4[10k+10-25]-6[3k+3+20]+2\times 55=0$

$40k-60-18k+110-138=0$

$22k=138+60-110$

$\therefore k=4$