Question

Show that the four points $(0,-1,-1),(4,5,1),(3,9,4)$ and $(-4,4,4)$ are coplaner and find the equation of the common plane.

Answer 1

The equation of the plane passing through the points $(0,-1,-1),(4,5,1)$ and $(3,9,4)$ is given by

$\left|\begin{array}{ccc}x-0& y+1& z+1\\ 4-0& 5+1& 1+1\\ 3-0& 9+1& 4+1\end{array}\right|=0$

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

$\Rightarrow$ $x(30-20)-(y+1)(20-6)+(z+1)(40-18)=0$

$\Rightarrow$ $10x-14(y+1)+22(z+1)=0$

$\Rightarrow$ $5x-7(y+1)+11(z+1)=0$

$\Rightarrow$ $5x-7y+11z+4=0$

Substituting the last points $(-4,4,4)$ i.e. $x=-4,y=4$ and $z=4$ in the above equation, we get,

$\Rightarrow$ $5(-4)-7\left(4\right)+11\left(4\right)+4=0$

$\Rightarrow$ $-48+48=0$

$\Rightarrow$ $0=0$

So, the plane equation is satisfied by the point $(-4,4,4).$

So, the given points are coplanar and the equation of the common plane is $5x-7y+11z+4=0$