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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Three Dimensional Geometry

NDA 2024 · Mathematics · Three Dimensional Geometry

Paragraph: Let S be the line of intersection of two planes x+y+z=1...

Paragraph: Let S be the line of intersection of two planes x+y+z=1x+y+z=1 and 2x+3y−4z=82 x+3 y-4 z=8.
Question: If (l,m,n)(l, m, n) are direction cosines of S, then what is the value of 43(l2−m2−n2)?43 \left(l²-m²-n²\right) ?

Options

  1. A66
  2. B55
  3. C44
  4. D11

Correct answer

A. 66

Step-by-step solution

2 planes are x+y+z=1and2x+3y−4z=8x+y+z=1 and 2 x+3 y-4 z=8. DC 's of line: S are 1,m,n1, \mathrm{ m}, \mathrm{n}1=−786,m=686,n=18643(12−m2−n2)=43(4986−3686−186)=43×1286=6\begin{aligned} & 1=\frac{-7}{\sqrt{86}}, m=\frac{6}{\sqrt{86}}, n=\frac{1}{\sqrt{86}} \\ & 43\left(1^2-m^2-n^2\right)=43\left(\frac{49}{86}-\frac{36}{86}-\frac{1}{86}\right)=43 \times \frac{12}{86}=6 \end{aligned}$$
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