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JEE Main 201322 AprilOnline
Exam
JEE Main
Year
2013
Date
22 April
Mode
Online
Subject
Mathematics
Chapter
Straight Lines

JEE Main 2013 · 22 April · Online · Mathematics · Straight Lines

The acute angle between two lines such that the direction cosines...

The acute angle between two lines such that the direction cosines l,m,nl, m, n, of each of them satisfy the equations l+m+n=0l+m+n=0 and l2+m2−n2=0l^2+m^2-n^2=0 is :

Options

  1. A15∘15^{\circ}
  2. B30∘30^{\circ}
  3. C60∘60^{\circ}
  4. D45∘45^{\circ}

Correct answer

C. 60∘60^{\circ}

Step-by-step solution

Let l1,m1,n1andl_1, m_1, n_1 andl_2, m_2, n_2bethed.cofline1and2respectively,thenasgivenbe the d.c of line 1 and 2 respectively, then as givenl_1+m_1+n_1=0 and l2+m2+n2=0l_2+m_2+n_2=0 and l12+m12−n12=0l_1{ }^2+m_1{ }^2-n_1^2=0 and l22+m22−n22=0(\becausel+m+n=0andl2+m2−n2=0)l_2^2+m_2^2-n_2^2=0 \left(\becausel+m+n=0\right. and \left.l^2+m²-n²=0\right) Angle between lines, \theta is \cos θ=\theta= l_1 l_2+m_1 m_2+n_1 n_2 As given l2+m2=n2l^2+m²=n² and l+m=−n⇒(−n)2−2lm=n2⇒2lm=0l+m=-n\Rightarrow(-n)^2-2 l m=n² \Rightarrow 2 l m=0 or lm=0l m=0 So l1m1=0,l2m2=0l_1 m_1=0, l_2 m_2=0 If l1=0,m1≠0l_1=0, m_1 ≠ 0 then l1m2=0l_1 m_2=0 If m1=0,l1≠0m_1=0, l_1 ≠ 0 then l2m1=0l_2 m_1=0 If l2=0,m2≠0l_2=0, m_2 ≠ 0 then l2m1=0l_2 m_1=0 If m2=0,l2≠0m_2=0, l_2 ≠ 0 then l1m2=0l_1 m_2=0 Also l1l2=0l_1 l_2=0 and m1m2=0m_1 m_2=0l2+m2−n2=l2+m2+n2−2n2=0⇒1−2n2=0⇒n=±12∴n1=±12,n2=±12∴cos⁡θ=12θ=60∘ (acute angle) \begin{aligned} & l²+m²-n²=l²+m²+n²-2 n²=0 \\ & \Rightarrow 1-2 n²=0 \Rightarrow n=\pm \frac{1}{\sqrt{2}}\\ & \therefore n_1=\pm \frac{1}{\sqrt{2}}, n_2=\pm \frac{1}{\sqrt{2}}\\ & \therefore \cos \theta=\frac{1}{2} \theta=60^{{}^{\circ}} \text{ (acute angle) } \end{aligned}$$
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