Question

A tetrahedron is a three dimensional figure bounded by non coplanar triangular planes. So a tetrahedron has four non-coplanar points as its vertices. Suppose a tetrehedron has points $A,B,C,D$ as it vertices which have coordinates $(x1,y1,z_1),(x_2,y_2,z_2)$ , $(x_3,y_3,z_3)$ and $(x_4,y_4,z_4)$ respectively in a rectangular three dimensional space. Then the coordinates of its centroid are $[\frac{x_1+x_2+x_3+x_4}{4},\frac{y_1+y_2+y_3+y_4}{4},\frac{z_1+z_2+z_3+z_4}{4}]$. Let $A$ tetrahedron has three of its vertices represented by the points $(0,0,0),(6,5,1)$ and $(4,1,3)$ and its centroid lies at the point $(1,2,5)$. Now answer the following two questions. The equation of the traingular plane of tetrahedron that contains the given vertices

• A. $x-2y+z=0$
• B. $5x-3y+2z=0$
• C. $-x+y+z=0$
• D. $x+2y+3z=0$

Answer 1

The correct option is C $-x+y+z=0$

Given three vertices of tetrahedron are $(0,0,0),(6,5,1),(4,1,3)$
Now equation of plane passing through $(0,0,0)$ is given by,
$ax+by+cz=0$
Also this plane passing through other two points
$\Rightarrow 6a+5b+c=0$ and $4a+b+3c=0$
Solving these, we get $a=-c,b=c$
Hence required plane is, $-x+y+z=0$