Three lines L _ 1 : → r = λ ^ i , λ ∈ R, L _ 2 : → r = ^ k + μ ^ j , μ ∈ R and L _ 3 : → r = ^ i + ^ j + ν ^ k , ν ∈ R are given. For which point(s) Q and L _ 2 can we find a point P on L _ 1 and a point R on L _ 3 so

Correct answer: C. ^ k + (1)/(2) ^ j As given L _ 1 ⇒ → r = , λ ∈ R L _ 2 ⇒ → r = k + , μ ∈ R and L _ 3 ⇒ → r = i + j + νk, ν ∈ R Let point P(λ, 0, 0), Q(0, μ, 1) and R(1, 1, ν) and → PQ = - λ ^ i + μ ^ j + ^ k , → PR = (1 - λ) ^ i + ^ j + ν ^ k as P, Q and R are co-linear ⇒ → PQ is parallel to → PR ⇒ - (λ)/(1 - λ) = (μ)/(1) = (1)/(ν) ⇒ λ = (μ)/(μ - 1) and ν = (1)/(μ) hence cases μ ≠ 0 & μ ≠ 1 cases Q ≠ (0, 0, 1) and Q ≠ (0, 1, 1) Q ≠ ^ k and Q ≠ ( ^ j + ^ k )

JEE Advanced 2019Paper 2

Exam: JEE Advanced
Year: 2019
Paper: Paper 2
Subject: Mathematics
Chapter: Three Dimensional Geometry

JEE Advanced 2019 · Paper 2 · Mathematics · Three Dimensional Geometry

Three lines L _ 1 : → r = λ ^ i , λ ∈ R, L _ 2 : → r = ^ k + μ ^ j...

Three lines

{L}_{1}: \overset{\to}{r} = \lambda\overset{^}{i}, \lambda \in R,

{L}_{2}: \overset{\to}{r} = \overset{^}{k} + \mu\overset{^}{j}, \mu \in R

and

{L}_{3}: \overset{\to}{r} = \overset{^}{i} + \overset{^}{j} + ν\overset{^}{k}, ν \in R

are given. For which point(s)

Q

and

{L}_{2}

can we find a point

P

{L}_{1}

and a point

R

{L}_{3}

so that

P, Q

and

R

are collinear?

Options

A $\overset{^}{k} + \overset{^}{j}$
B $\overset{^}{k}$
C $\overset{^}{k} + \dfrac{1}{2}\overset{^}{j}$
D $\overset{^}{k} - \dfrac{1}{2}\overset{^}{j}$

Correct answer

C. $\overset{^}{k} + \dfrac{1}{2}\overset{^}{j}$

Step-by-step solution

As given

{L}_{1} ⇒ \overset{\to}{r} = \lambdai, \lambda \in R

{L}_{2} ⇒ \overset{\to}{r} = k + \muj, \mu \in R

and

{L}_{3} ⇒ \overset{\to}{r} = i + j + νk, ν \in R

Let point

P\left(\lambda, 0, 0\right), Q\left(0, \mu, 1\right)

and

R\left(1, 1, ν\right)

and

\overset{\to}{PQ} = - \lambda\overset{^}{i} + \mu\overset{^}{j} + \overset{^}{k}, \overset{\to}{PR} = \left(1 - \lambda\right)\overset{^}{i} + \overset{^}{j} + ν\overset{^}{k}

P, Q

and

R

are co-linear

⇒ \overset{\to}{PQ}

is parallel to

\overset{\to}{PR}

⇒ - \dfrac{\lambda}{1 - \lambda} = \dfrac{\mu}{1} = \dfrac{1}{ν}

⇒ \lambda = \dfrac{\mu}{\mu - 1}

and

ν = \dfrac{1}{\mu}

hence

\begin{cases}\mu \ne 0 & \mu \ne 1\end{cases}

Q \ne \left(0, 0, 1\right)

and

Q \ne \left(0, 1, 1\right)

Q \ne \overset{^}{k} \text{ and } Q \ne \left(\overset{^}{j} + \overset{^}{k}\right)

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