A point on a line has coordinates (p + 1, p - 3, √2p) where p is any real number. What are the direction cosines of the line?

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  1. \(\frac{1}{2},\frac{1}{2},\frac{1}{\sqrt{2}}\)
  2. \(\frac{1}{\sqrt{2}},\frac{1}{2},\frac{1}{2}\)
  3. \(\frac{1}{\sqrt{2}},\frac{1}{2},-\frac{1}{2}\)
  4. Cannot be determined due to insufficient data

Answer (Detailed Solution Below)

Option 1 : \(\frac{1}{2},\frac{1}{2},\frac{1}{\sqrt{2}}\)
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Detailed Solution

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Concept:

If a, b and c are direction ratio’s of a line then direction cosines of the line are given by:

\(l=\frac{a}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}},~m=~\frac{b}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}~and~n=~\frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\)

Calculation:

Given: A point on a line has coordinates (p + 1, p - 3, √2p)

⇒ x = p + 1 ⇒ x - 1 = p     ---(1)

⇒ y = p - 3 ⇒ y + 3 = p     ---(2)

⇒ z = √2 × p ⇒ \(\frac{z}{\sqrt{2}}\)  = p   ---(3)

From (1), (2) and (3) we can say that

\(\frac{x-1}{1}=\frac{y+3}{1}=\frac{z-0}{\sqrt{2}}=p\)

∴ Direction ratios are: 1, 1, √2

⇒ Direction cosines are: \(\frac{1}{2},\frac{1}{2},\frac{1}{\sqrt{2}}\)
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