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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Three Dimensional Geometry

NDA 2023 · Mathematics · Three Dimensional Geometry

Paragraph: Consider two lines whose direction ratios are (2,-1,2)...

Paragraph: Consider two lines whose direction ratios are (2,−1,2)(2,-1,2) and (k,3,5)(k, 3,5). They are inclined at an angle π4\frac{\pi}{4}.
Question: What is the value of kk ?

Options

  1. A44
  2. B22
  3. C11
  4. D−1-1

Correct answer

A. 44

Step-by-step solution

A(2,−1,2)B(k,3,5)cos⁡θ=2k−3+1022+(−1)2+22⋅k2+32+52⇒cos⁡π4=2k+73⋅34+k2=12⇒334+k2=2(2k+7)⇒9(34+k2)=2(2k+7)2⇒306+9k2=8k2+98+56k⇒k2−56k+208=0⇒(k−52)(k−4)=0⇒k=4,52∴k=4A(2,-1,2) B(k, 3,5)\cos \theta=\frac{2 k-3+10}{\sqrt{2^2+(-1)^2+2^2} \cdot \sqrt{k^2+3^2+5^2}}\Rightarrow \quad \cos \frac{\pi}{4}=\frac{2 k+7}{3 \cdot \sqrt{34+k^2}}=\frac{1}{\sqrt{2}}\Rightarrow \quad 3 \sqrt{34+k^2}=\sqrt{2}(2 k+7)\Rightarrow \quad 9\left(34+k^2\right)=2(2 k+7)^2\Rightarrow \quad 306+9 k^2=8 k^2+98+56 k\Rightarrow \quad k^2-56 k+208=0\Rightarrow \quad(k-52)(k-4)=0\Rightarrow \quad k=4,52\therefore \quad k=4
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