If a, b, c are the direction ratios of the line and l, m, n are the direction cosines of the line, then which of the following is incorrect?

Category: QuestionsIf a, b, c are the direction ratios of the line and l, m, n are the direction cosines of the line, then which of the following is incorrect?
Editor">Editor Staff asked 11 months ago

If a, b, c are the direction ratios of the line and l, m, n are the direction cosines of the line, then which of the following is incorrect?
 
(a) \(\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=k\)
 
(b) l^2+m^2+n^2=1
 
(c) k=±\(\frac{1}{\sqrt{(a^2+b^2+c^2)}}\)
 
(d) l^2-m^2=n^2-1
 
This question was posed to me during a job interview.
 
The query is from Direction Cosines and Direction Ratios of a Line in division Three Dimensional Geometry of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
Interview Questions and Answers, Database Interview Questions and Answers for Freshers and Experience

1 Answers
Editor">Editor Staff answered 11 months ago

The correct option is (d) l^2-m^2=n^2-1
 
Explanation: Given that, a, b, c are the direction ratios of the line and l, m, n are the direction cosines of the line,
 
\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=k and l^2+m^2+n^2=1
 
⇒l=ak, m=bk, n=ck
 
(ak)^2+(bk)^2+(ck)^2=1
 
k^2 (a^2+b^2+c^2)=1
 
k^2=\frac{1}{a^2+b^2+c^2}
 
∴k=±\frac{1}{\sqrt{(a^2+b^2+c^2)}}
 
Hence, l^2-m^2=n^2-1 is incorrect.