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Q.
The volume of the tetrahedron (in cubic units) formed by the plane $2 x+y+z=K$ and the coordinate planes is $\frac{2 V^{3}}{3}$, then $K: V=$
TS EAMCET 2019
Solution:
Equation of given plane is
$2 x+y+z=K$
Point of intersection of plane (i) with the coordinate axes is $A\left(\frac{K}{2}, 0,0\right), B(0, K, 0)$ and $C(0,0, K)$.
Now, volume of the tetrahedron $O A B C$
$=\frac{1}{6}[ O A O B O C ]=\frac{1}{6}\begin{vmatrix}\frac{K}{2} & 0 & 0 \\ 0 & K & 0 \\ 0 & 0 & K\end{vmatrix}=\frac{2 V^{3}}{3}($ given $)$
$\Rightarrow \frac{1}{6} \frac{K^{3}}{2}=\frac{2 V^{3}}{3}$
$\Rightarrow \left(\frac{K}{V}\right)^{3}=2^{3}$
$ \Rightarrow K: V=2: 1$