If the direction cosines of a line ‘L’ are (ab,b,b) and the angle between ‘L’ and x-axis is 'π3 . Then a pair of possible values for a,b are

(l,m,n)=(ab,b,b)∴ab=cos⁡π3=12⇒ab=12(ab)2+b2+b2=1⇒2b2=1−14=34⇒b=38⇒ but ab=12⇒a=23

If the direction cosines of a line ‘L’ are (ab,b,b) and the angle between ‘L’ and x-axis is 'π3 . Then a pair of possible values for a,b are

(l,m,n)=(ab,b,b)∴ab=cos⁡π3=12⇒ab=12(ab)2+b2+b2=1⇒2b2=1−14=34⇒b=38⇒ but ab=12⇒a=23

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If the direction cosines of a line ‘L’ are (ab,b,b) and the angle between ‘L’ and x-axis is $\frac{^{'} π}{3}$ . Then a pair of possible values for a,b are

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$\sqrt{\frac{2}{3}}, \sqrt{\frac{3}{8}}$

$\sqrt{\frac{8}{3}}, \sqrt{\frac{2}{3}}$

2,5

3,4

answer is A.

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Detailed Solution

$\begin{array}{l} (l, m, n) = (ab, b, b) \\ ∴ ab = \cos \frac{π}{3} = \frac{1}{2} \Rightarrow ab = \frac{1}{2} \\ (ab)^{2} + b^{2} + b^{2} = 1 \Rightarrow 2 b^{2} = 1 - \frac{1}{4} = \frac{3}{4} \\ \Rightarrow b = \sqrt{\frac{3}{8}} \Rightarrow but ab = \frac{1}{2} \Rightarrow a = \sqrt{\frac{2}{3}} \end{array}$