Let the image of the point $\mathrm{P}(1,2,3)$ in the line $\mathrm{L}: \frac{\mathrm{x}-6}{3}=\frac{\mathrm{y}-1}{2}=\frac{\mathrm{z}-2}{3}$ be Q. Let $\mathrm{R}(\alpha, \beta, \gamma)$ be a point that divides internally the line segment PQ in the ratio $1: 3$. Then the value of $22(\alpha+\beta+\gamma)$ is equal to

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

Let the image of the point

P (1, 2, 3)

in the line

L : \frac{x - 6}{3} = \frac{y - 1}{2} = \frac{z - 2}{3}

be Q. Let

R (α, β, γ)

be a point that divides internally the line segment PQ in the ratio

1 : 3

. Then the value of

22 (α + β + γ)

is equal to

Answer: 125

Let M be the foot of perpendicular from P to line L. Point M on L is

(3 λ + 6, 2 λ + 1, 3 λ + 2)

. Vector

P M = (3 λ + 5) \hat{i} + (2 λ - 1) \hat{j} + (3 λ - 1) \hat{k}

is perpendicular to the direction vector

(3, 2, 3)

. So

3 (3 λ + 5) + 2 (2 λ - 1) + 3 (3 λ - 1) = 0

, giving

λ = - \frac{5}{11}

. Thus

M (\frac{51}{11}, \frac{1}{11}, \frac{7}{11})

. Since M is the midpoint of PQ, we find Q. Point R divides PM in ratio 1:3, so R is the midpoint of PM. Computing:

α = \frac{1 + \frac{51}{11}}{2} = \frac{31}{11}

β = \frac{2 + \frac{1}{11}}{2} = \frac{23}{22}

γ = \frac{3 + \frac{7}{11}}{2} = \frac{19}{11}

. Therefore,

22 (α + β + γ) = 22 (\frac{31}{11} + \frac{23}{22} + \frac{19}{11}) = 125

JEE Main · 2022 · 28 Jun 2022 · Shift 2 · Integer answer