Let a plane P pass through the point $(3,7,-7)$ and contain the line, $\frac{\mathrm{x}-2}{-3}=\frac{\mathrm{y}-3}{2}=\frac{\mathrm{z}+2}{1}$. If distance of the plane $P$ from the origin is $d$, then $d^{2}$ is equal to \_\_\_\_ .

JEE Main · 2021 · 27 Jul 2021 · Shift 1 · Three Dimensional Geometry

Let a plane P pass through the point

(3, 7, - 7)

and contain the line,

\frac{x - 2}{- 3} = \frac{y - 3}{2} = \frac{z + 2}{1}

. If distance of the plane

P

from the origin is

d

, then

d^{2}

is equal to ____ .

Answer: 3

Solution

The line passes through

A (2, 3, - 2)

with direction

b = (- 3, 2, 1)

, and

B (3, 7, - 7)

lies on the plane, with

A B = (1, 4, - 5)

. The normal is

n = b \times A B = (- 14, - 14, - 14)

so the plane has the form

x + y + z = k

; the point

(3, 7, - 7)

gives

k = 3

, and

A

satisfies it as well, since

2 + 3 - 2 = 3

. The plane is

x + y + z = 3

.
Its distance from the origin is

d = \frac{3}{3} = 3

, so

d^{2} = 3

JEE Main · 2021 · 27 Jul 2021 · Shift 1 · Integer answer

Practise more: all JEE Main PYQs on Three Dimensional Geometry →

More JEE Main questions on Three Dimensional Geometry

Let $(α, β, γ)$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5, 4, 2)$ on the line $r = (- \hat{i} + 3 \hat{j} + \hat{k}) + λ (2 \hat{i} + 3 \hat{j} - \hat{k})$ .
Then the length of the projection of the vector …
JEE Main 2026View solution →
Let the line L pass through the point $(- 3, 5, 2)$ and make equal angles with the positive coordinate axes. If the distance of L from the point $(- 2, r, 1)$ is $\frac{14}{3}$ , then the sum of all possible values of $r$ is :
JEE Main 2026View solution →
Let the line $L_{1}$ be parallel to the vector $- 3 \hat{i} + 2 \hat{j} + 4 \hat{k}$ and pass through the point $(2, 6, 7)$ and the line $L_{2}$ be parallel to the vector $2 \hat{i} + \hat{j} + 3 \hat{k}$ and pass through the point $(4, 3, 5)$ . If the line $L_{3}$ is parallel to the vector …
JEE Main 2026View solution →
Let $P (α, β, γ)$ be the point on the line
$\frac{x - 1}{2} = \frac{y + 1}{- 3} = z$ at a distance $414$ from the point $(1, - 1, 0)$ and nearer to the origin. Then the shortest distance, between the lines …
JEE Main 2026View solution →
If the image of the point $P (1, 2, a)$ in the line $\frac{x - 6}{3} = \frac{y - 7}{2} = \frac{7 - z}{2}$ is $Q (5, b, c)$ , then $a^{2} + b^{2} + c^{2}$ is equal to
JEE Main 2026View solution →
Let L be the line $\frac{x + 1}{2} = \frac{y + 1}{3} = \frac{z + 3}{6}$ and let S be the set of all points $(a, b, c)$ on L, whose distance from the line $\frac{x + 1}{2} = \frac{y + 1}{3} = \frac{z - 9}{0}$ along the line L is 7. Then $\sum_{(a, b, c) \in S} (a + b + c)$ is equal to :
JEE Main 2026View solution →