Question

Find x such that the four points A(4, 4, 4), B(5, x, 8), C(5, 4, 1) and D(7, 7, 2) are coplanar.

Answer 1

Equation of plane formed by the points $A,C,D$ is given by :

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

$(x-4)\left(9\right)-(y-4)\left(7\right)+3(z-4)=0$

$9x-36-7y+28+3z-12=0$

$9x-7y+3z-20=0$

Since, the points are coplanar so point B lies on the plane.

Therefore,

$45-7x+24-20=0$

$7x=49$

$x=7$