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  4. A plane P is parallel to two lines whose direction ratios are -2,1,-3, and -1,2,-2 and it contains the point (2,2,-2). Let P intersect the co-ordinate axes at the points A, B , C making the intercepts α, β, γ. If V is the volume of the tetrahedron OABC, where O is the origin and p=α+β+γ, then the ordered pair ( V , p ) is equal to

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Q. A plane $P$ is parallel to two lines whose direction ratios are $-2,1,-3$, and $-1,2,-2$ and it contains the point $(2,2,-2)$. Let $P$ intersect the co-ordinate axes at the points $A$, $B , C$ making the intercepts $\alpha, \beta, \gamma$. If $V$ is the volume of the tetrahedron $OABC$, where $O$ is the origin and $p=\alpha+\beta+\gamma$, then the ordered pair $( V , p )$ is equal to

JEE MainJEE Main 2022Three Dimensional Geometry

Solution:

Normal of plane $P$ :
$=\begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\-2 & 1 & -3 \\-1 & 2 & -2\end{vmatrix}=4 \hat{i}-\hat{j}-3 \hat{k}$
Equation of plane $P$ which passes through (2, $2,-2)$ is $4 x-y-3 z-12=0$
Now, A $(3,0,0)$, B $(0,-12,0)$, C $(0,0,-4)$
$ \Rightarrow \alpha=3, \beta=-12, \gamma=-4$
$\Rightarrow p =\alpha+\beta+\gamma=-13$
Now, volume of tetrahedron $OABC$
$ V =\left|\frac{1}{6} \overrightarrow{ OA } \cdot(\overrightarrow{ OB } \times \overrightarrow{ OC })\right|=24$
$( V , p )=(24,-13)$\