question_answer

If the plane \[\frac{x}{2}+\frac{y}{3}+\frac{z}{4}=\frac{1}{2}\] intersects x, y and z-axes at A, B and C, respectively, then volume of the tetrahedron OABC where O is the origin is

A) \[1/3\]                    

B)        \[1/2\]        

C)    \[2\]                       

D)       \[3\]

Correct Answer: B

Solution :

  [b] Given plane is  \[\frac{x}{2}+\frac{y}{3}+\frac{z}{4}=\frac{1}{2}\] From the figure, volume of tetrahedron, \[V=\frac{1}{6}\left[ \overrightarrow{OA}\,\,\overrightarrow{OB}\,\,\overrightarrow{OC} \right]\] \[=\frac{1}{6}\left[ \hat{i}\,\,\,\frac{3}{2}\hat{j}\,\,2\hat{k} \right]=\frac{1}{6}\times \frac{3}{2}\times 2=\frac{1}{2}\]