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  • The foot of perpendicular from the origin to a plane <img alt="\mathrm{P}" src="https://entrancecorne

The foot of perpendicular from the origin \mathrm{O} to a plane \mathrm{P} which meets the co-ordinate axes at the points \mathrm{ A, B, C \: is \: (2, a, 4), a \in N}. If the volume of the tetrahedron \mathrm{OABC} is 144 \mathrm{unit}^3, then which of the following points is NOT on \mathrm{P} ?

 

Option: 1

(0,6,3)
 


Option: 2

(0,4,4)
 


Option: 3

(2,2,4)
 


Option: 4

(3,0,4)


Answers (1)

\mathrm{\vec{n}=(2, a, 4)}
Plane is
\mathrm{ 2 x+a y+4 z=4+\mathrm{a}^2+16 }
\mathrm{ =20+\mathrm{a}^2 }
\mathrm{ \mathrm{~A}\left(\frac{20+\mathrm{a}^2}{2}, 0,0\right) }
\mathrm{ \mathrm{B}\left(0, \frac{20+\mathrm{a}^2}{\mathrm{a}}, 0\right)}

\mathrm{C\left(0,0, \frac{20+a^2}{4}\right) }
\mathrm{ \frac{1}{6} \times \frac{\left(20+a^2\right)^3}{8 a}=144=2^4 \times 3^2 }

\mathrm{ \left(20+a^2\right)^3=2^8 3^3 a }

\mathrm{ 20+a^2=(4 a)^{\frac{1}{3}}(12)}

\mathrm{a=2} satisfies above equation

So, \mathrm{2 x+2 y+4 z=24}

\mathrm{ \mathrm{X}+\mathrm{Y}+2 \mathrm{z}=12 }

\mathrm{(A) (0,6,3) }

\mathrm{(B) (0,4,4) }

\mathrm{(C) (2,2,4) }

\mathrm{(D) (3,0,4) }

 

Posted by

Sumit Saini

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