3D Geometry Ques 74
74. A plane is parallel to two lines whose direction ratios are $(1,0,-1)$ and $(-1,1,0)$ and it contains the point $(1,1,1)$. If it cuts coordinate axes at $A, B, C$. Then find the volume of the tetrahedron $O A B C$.
(2004, 2M)
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Answer:
Correct Answer: 74.$\frac{9}{2}$ cu units Solution:
Formula:
- Let the equation of plane through $(1,1,1)$ having $a, b, c$ as DR’s of normal to plane,
$ a(x-1)+b(y-1)+c(z-1)=0 $
and plane is parallel to straight line having DR’s.
$ (1,0,-1) \text { and }(-1,1,0) $
$\Rightarrow$
$ \begin{aligned} a-c & =0 \\ -a+b & =0 \\ a=b & =c \end{aligned} $
$\Rightarrow$
$\therefore$ Equation of plane is
$ x-1+y-1+z-1=0 \quad \text { or } \quad \frac{x}{3}+\frac{y}{3}+\frac{z}{3}=1 \text {. } $
Its intercept on coordinate axes are
$ A(3,0,0), B(0,3,0), C(0,0,3) \text {. } $
Hence, the volume of tetrahedron $O A B C$
$ =\frac{1}{6}[\mathbf{a} \mathbf{b} \mathbf{c} \mathbf{c}]=\frac{1}{6}\left|\begin{array}{lll} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{array}\right|=\frac{27}{6}=\frac{9}{2} \text { cu units } $
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