Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

Equation of the plane passing through $(1,1,1)$ is given as $$\mathrm{a}(x-1)+\mathrm{b}(y-1)+\mathrm{c}(\mathrm{z}-1)=0$$ As the plane is parallel to the lines having direction ratios $1,0,-1$ and $-1,1,0$, we get $$\begin{array}{rl}& a-c=0\text{ and }-a+b=0\\ & \Rightarrow a=b=c\end{array}$$ $\therefore$ From (i) and (ii), we get $$\begin{array}{rl}& \therefore x-1+y-1+z-1=0\\ & \therefore x+y+z=3\Rightarrow \frac{x}{3}+\frac{y}{3}+\frac{z}{3}=1\end{array}$$ $\therefore$ Co-ordinates of A, B, C are $(3,0,0),(0,3,0)$ and $(0,0,3)$ respectively. $$\begin{array}{rl}\therefore \text{ Volume of tetrahedron OABC }& =\frac{1}{6}\left|\begin{array}{lll}3& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right|\\ & =\frac{1}{6}\times 27\\ & =\frac{9}{2}\text{ cu. units }\end{array}$$