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

JEE Main · 2023 · 31 Jan 2023 · Shift 2 · Three Dimensional Geometry

The foot of perpendicular from the origin O to a plane P which meets the co-ordinate axes at the points

A, B, C

(2, a, 4), a \in N

. If the volume of the tetrahedron OABC is 144 unit

^{3}

, then which of the following points is NOT on P ?

Answer: (C)

The foot of the perpendicular

N (2, a, 4)

gives the normal direction, so the plane is

2 x + a y + 4 z = ∣ O N ∣^{2} = 20 + a^{2} .

Its axis intercepts are

\frac{20 + a ^{2}}{2}, \frac{20 + a ^{2}}{a}, \frac{20 + a ^{2}}{4}

, so the tetrahedron

O A B C

has volume

\frac{( 20 + a ^{2} ) ^{3}}{48 a} .

Setting this equal to

144

gives

(20 + a^{2})^{3} = 6912 a

, satisfied by

a = 2

, and the plane becomes

x + y + 2 z = 12.

Testing the points,

(3, 0, 4)

gives

3 + 0 + 8 = 11 \neq = 12

, so it does not lie on the plane.

JEE Main · 2023 · 31 Jan 2023 · Shift 2 · Single correct option