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

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The query is from Direction Cosines and Direction Ratios of a Line in division Three Dimensional Geometry of Mathematics – Class 12

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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.