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← JEE Advanced 2019 · Paper 2
MathematicsMultiple correct+4 / -1Question 44 of 54

JEE Advanced 2019 · Paper 2, Mathematics question 8

Three lines

L1: r⃗=λi^, λ∈R,L_1:\ \vec r = \lambda\hat i,\ \lambda\in\mathbb R,

L2: r⃗=k^+μj^, μ∈RandL_2:\ \vec r = \hat k + \mu\hat j,\ \mu\in\mathbb R\quad\text{and}

L3: r⃗=i^+j^+vk^, v∈RL_3:\ \vec r = \hat i + \hat j + v\hat k,\ v\in\mathbb R

are given. For which point(s) QQ on L2L_2 can we find a point PP on L1L_1 and a point RR on L3L_3 so that PP, QQ and RR are collinear?

Choose one or more options
Answer and worked solution

Official answer: A, C

Take P=(λ,0,0)P=(\lambda,0,0), Q=(0,μ,1)Q=(0,\mu,1) and R=(1,1,v)R=(1,1,v).

Collinearity. Require R−Q=t(P−Q)R-Q=t(P-Q):
(1, 1−μ, v−1)=t(λ, −μ, −1).(1,\,1-\mu,\,v-1)=t(\lambda,\,-\mu,\,-1).

  • The second component gives t=μ−1μt=\dfrac{\mu-1}\mu, so μ≠0\mu\ne0.
  • The first gives tλ=1t\lambda=1, so t≠0t\ne0, i.e. μ≠1\mu\ne1.
  • λ\lambda and vv then follow.

Which Q work. Any μ≠0,1\mu\ne0,1 works:

  • (A) μ=−12\mu=-\frac12 ✓
  • (B) μ=0\mu=0 ✗
  • (C) μ=12\mu=\frac12 ✓
  • (D) μ=1\mu=1 ✗

(A), (C)

Topics:Equation of a line in spaceVector operations
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