Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2019
Exam
NDA
Year
2019
Subject
Mathematics
Chapter
Three Dimensional Geometry

NDA 2019 · Mathematics · Three Dimensional Geometry

A straight line passes through the point (1,1,1) makes an angle 60^...

A straight line passes through the point (1,1,1)(1,1,1) makes an angle 60∘60^{{}^{\circ}} with the positive direction of z -axis, and the cosine of the angles made by it with the positive directions of the yy-axis and the xx-axis are in the ratio 3\sqrt{3}: 1.Whatistheacuteanglebetweenthetwopossiblepositionsoftheline?. What is the acute angle between the two possible positions of the line?

Options

  1. A90∘90^{{}^{\circ}}
  2. B60∘60^{{}^{\circ}}
  3. C45∘45^{{}^{\circ}}
  4. D30∘30^{{}^{\circ}}

Correct answer

B. 60∘60^{{}^{\circ}}

Step-by-step solution

given angle from zz-axis =60∘∴n=cos⁡60∘=12=60^{{}^{\circ}}\therefore \quad n=\cos 60^{{}^{\circ}}=\frac{1}{2} Let the cosines of angle made by line with yy-axis and x-axis be 3x and x∵l2+m2+n2=1⇒3x2+x2+14=1⇒x=±34∴3x=34x=34\begin{aligned} & x \text{-axis be } \sqrt{3} x \text{ and } x \\ & \because \quad l^2+m^2+n^2=1 \\ & \Rightarrow 3 x^2+x^2+\frac{1}{4}=1 \\ & \Rightarrow \quad x= \pm \frac{\sqrt{3}}{4} \\ & \therefore \quad \sqrt{3} x=\frac{3}{4} \\ & x=\frac{\sqrt{3}}{4} \end{aligned}∴(l,m,n)=(34,34,12)or (−34,−34,12)for angle ∣l1l2+m1m2+n1n2∣=∣−916−316+14∣=∣−12∣=cos⁡θ⇒ Angle =θ=60∘\begin{aligned} & \therefore \quad(l, m, n)=\left(\frac{3}{4}, \frac{\sqrt{3}}{4}, \frac{1}{2}\right) \\ & \text{or }\left(\frac{-3}{4}, \frac{-\sqrt{3}}{4}, \frac{1}{2}\right) \\ & \text{for angle } \\ & \left|l_1 l_2+m_1 m_2+n_1 n_2\right| \\ & =\left|\frac{-9}{16}-\frac{3}{16}+\frac{1}{4}\right| \\ & =\left|\frac{-1}{2}\right|=\cos \theta \\ & \Rightarrow \text{ Angle }=\theta=60^{{}^{\circ}}\end{aligned}$
Practice Three Dimensional Geometry on Quantrex Academy →

More from Three Dimensional Geometry

Passage: The equation of the sphere S is x^2+y^2+z^2-4x-6y-12z+k=0 . Question: If the… NDA 2026Passage: A line L passing through the point (-1,2,-3) is perpendicular to the plane P… NDA 2026Passage: P(-2,-3,5) , Q(4,-1,5) , R(6,-4,8) and S(2,-6,10) are four points. Question… NDA 2026Passage: A line L passing through the point (-1,2,-3) is perpendicular to the plane P… NDA 2026Passage: P(-2,-3,5) , Q(4,-1,5) , R(6,-4,8) and S(2,-6,10) are four points. Question… NDA 2026Passage: The equation of the sphere S is x^2+y^2+z^2-4x-6y-12z+k=0 . Question: What is… NDA 2026For the following two (02) items: A plane P is parallel to the line having direction… NDA 2025For the following two (02) items: Suppose S is the sphere with the smallest radius that… NDA 2025A(1, 2, -1) , B(2, 5, -2) and C(4, 4, -3) are three vertices of a rectangle. What is the… NDA 2025If a line (x+1)/(p) = (y-1)/(q) = (z-2)/(r) where p = 2q = 3r , makes an angle θ with the… NDA 2025What is the equation of the plane passing through the point (1, 1, 1) and perpendicular… NDA 2025For the following two (02) items: Let A(1, -1, 0) , B(-2, 1, 8) and C(-1, 2, 7) are three… NDA 2025For the following two (02) items: Suppose S is the sphere with the smallest radius that… NDA 2025If a line in 3 dimensions makes angles α , β and γ with the positive directions of the… NDA 2025A line makes angles α , β and γ with the positive directions of the coordinate axes. If a… NDA 2025For the following two (02) items: A plane P is parallel to the line having direction… NDA 2025 Full Three Dimensional Geometry list Three Dimensional Geometry JEE Maths PYQsJEE Mathematics PYQs All NDA PYQs