If the line $\frac{2-x}{3}=\frac{3y-2}{4\lambda+1}=4-z$ makes a right angle with the line $\frac{x+3}{3\mu}=\frac{1-2y}{6}=\frac{5-z}{7}$, then $4\lambda+9\mu$ is equal to:

JEE Main · 2024 · 05 Apr 2024 · Shift 1 · Three Dimensional Geometry

If the line

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

makes a right angle with the line

\frac{x + 3}{3 μ} = \frac{1 - 2 y}{6} = \frac{5 - z}{7}

, then

4 λ + 9 μ

is equal to:

(A)4
(B)6✓ Correct
(C)13
(D)5

Answer: (B)

Solution

Direction ratios: first line

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

gives

(- 3, \frac{4 λ + 1}{3}, - 1)

; second line

\frac{x + 3}{3 μ} = \frac{1 - 2 y}{6} = \frac{5 - z}{7}

gives

(3 μ, - 3, - 7)

.
Perpendicular

\Rightarrow

dot product

= 0

(- 3) (3 μ) + \frac{4 λ + 1}{3} (- 3) + (- 1) (- 7) = 0 \Rightarrow - 9 μ - (4 λ + 1) + 7 = 0

.
So

4 λ + 9 μ = 6.

JEE Main · 2024 · 05 Apr 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 →