If the image of the point $(4,4,3)$ in the line $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-1}{3}$ is $(\alpha, \beta, \gamma)$, then $\alpha+\beta+\gamma$ is equal to

JEE Main · 2025 · 28 Jan 2025 · Shift 1 · Three Dimensional Geometry

If the image of the point

(4, 4, 3)

in the line

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

(α, β, γ)

, then

α + β + γ

is equal to

(A)9✓ Correct
(B)12
(C)8
(D)7

Answer: (A)

Solution

A general point on the line is

Q (2 λ + 1, λ + 2, 3 λ + 1)

P Q = (2 λ - 3, λ - 2, 3 λ - 2)

, where

P = (4, 4, 3)

For

Q

to be the foot of the perpendicular,

P Q

must be perpendicular to the direction vector

(2, 1, 3)

2 (2 λ - 3) + 1 (λ - 2) + 3 (3 λ - 2) = 0 \Rightarrow 14 λ - 14 = 0 \Rightarrow λ = 1

So the foot of perpendicular is

Q (3, 3, 4)

.

Since

Q

is the midpoint of

P

and its image

R (α, β, γ)

α = 2 Q_{x} - P_{x} = 6 - 4 = 2, β = 2 Q_{y} - P_{y} = 6 - 4 = 2, γ = 2 Q_{z} - P_{z} = 8 - 3 = 5

α + β + γ = 2 + 2 + 5 = 9

JEE Main · 2025 · 28 Jan 2025 · Shift 1 · Single correct option

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 →