JEE Main · 2024 · 30 Jan 2024 · Shift 1 · Three Dimensional Geometry

Let

(α, β, γ)

be the foot of perpendicular from the point (1,2,3) on the line

\frac{x + 3}{5} = \frac{y - 1}{2} = \frac{z + 4}{3}

. Then

19 (α + β + γ)

is equal to :

(A)102
(B)101✓ Correct
(C)99
(D)100

Answer: (B)

Solution

Parametrise the line as

(5 k - 3, 2 k + 1, 3 k - 4)

. The vector from

(1, 2, 3)

to this point is

(5 k - 4, 2 k - 1, 3 k - 7)

, which must be perpendicular to the direction

(5, 2, 3)

5 (5 k - 4) + 2 (2 k - 1) + 3 (3 k - 7) = 0 \Rightarrow 38 k = 43 \Rightarrow k = \frac{43}{38}

α + β + γ = (5 k - 3) + (2 k + 1) + (3 k - 4) = 10 k - 6 = \frac{430}{38} - 6 = \frac{101}{19}

19 (α + β + γ) = 101

JEE Main · 2024 · 30 Jan 2024 · 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 →