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  • A plane  is parallel to two lines whose direction ratios are <img alt="\mathrm{-2,1,-3}" src="/latex-image/?%5Cmathrm%7B-2%2C1%2C-3

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

Option: 1

(48,-13)


Option: 2

(24,-13)


Option: 3

(48,11)


Option: 4

(24,-5)


Answers (1)

best_answer

\begin{aligned} \vec{n} &=\left|\begin{array}{ccc} \mathrm{i} &\mathrm{ j} & \mathrm{k} \\ \mathrm{-2} & \mathrm{1} & \mathrm{-3} \\ \mathrm{-1} & \mathrm{2} & \mathrm{-2} \end{array}\right| \\ &=\mathrm{4 i-j-3 k} \end{aligned} \\ \text{Equation of plarie is}\\ \begin{gathered} \mathrm{4(x-2)-1(y-2)-3(z+2)=0} \\ \mathrm{4 x-8-y+2-3 z-6=0} \\ \mathrm{4 x-y-3 z=12} \\ \mathrm{\frac{x}{3}+\frac{y}{-12}+\frac{z}{(-4)}=1} \\ \mathrm{\alpha=3, \beta=-12,-4} \\ \mathrm{\alpha+\beta+\gamma=-13} \end{gathered}\\ \begin{aligned} \text { Volume } &=\mathrm{\frac{1}{6}} \left[\begin{array}{ll} \mathrm{\alpha} &\mathrm{ \beta} \end{array}\right] \\ &=\mathrm{\frac{1}{6} \times 3 \times 12 \times-4} \\ &=\mathrm{24} \\ \therefore \text { option (B) } \end{aligned}

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SANGALDEEP SINGH

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