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 $\mathrm{A}, \mathrm{B}, \mathrm{C}$ making the intercepts $\alpha, \beta, \gamma$. If V is the volume of the tetrahedron OABC, where O is the origin and $\mathrm{p}=\alpha+\beta+\gamma$, then the ordered pair ($\mathrm{V}, \mathrm{p}$) is equal to

JEE Main · 2022 · 28 Jul 2022 · Shift 2 · Three Dimensional Geometry

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

Answer: (B)

The normal to plane P is perpendicular to both given direction vectors:

n = \hat{i} - 2 - 1 \hat{j} 12 \hat{k} - 3 - 2 = 4 \hat{i} - \hat{j} - 3 \hat{k}

. The plane equation passing through

(2, 2, - 2)

4 x - y - 3 z - 12 = 0

. The intercepts are

α = 3

(set

y = z = 0

β = - 12

(set

x = z = 0

γ = - 4

(set

x = y = 0

). Thus

p = 3 - 12 - 4 = - 13

. The volume of tetrahedron OABC is

V = \frac{1}{6} ∣ α β γ ∣ = \frac{1}{6} \times 3 \times 12 \times 4 = 24

. Therefore,

(V, p) = (24, - 13)

JEE Main · 2022 · 28 Jul 2022 · Shift 2 · Single correct option