If the direction cosines of a line L are (ab, b, b) and the Angle between L and x – axis is $\frac{\pi }{3}$ then the value of a² + b² is equal to

D c’s of L : (ab, b, b)

D c’s of x – axis : (1, 0, 0)

$\text{cos }\frac{\text{π}}{\text{3}}\text{=ab}$

$\Rightarrow \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}ab\hspace{0.17em}=\hspace{0.17em}}\frac{1}{2}$  and  ${\text{a}}^{\text{2}}{\text{b}}^{\text{2}}{\text{\hspace{0.17em}+\hspace{0.17em}b}}^{\text{2}}{\text{+b}}^{\text{2}}\text{\hspace{0.17em}=\hspace{0.17em}1}$

$\frac{\text{1}}{\text{4}}{\text{\hspace{0.17em}+\hspace{0.17em}2b}}^{\text{2}}\text{=1}$

$\Rightarrow {\text{\hspace{0.17em}\hspace{0.17em}2b}}^{\text{2}}\text{=}\frac{\text{3}}{\text{4}}\Rightarrow {\text{\hspace{0.17em}b}}^{\text{2}}\text{=}\frac{\text{3}}{\text{8}}$

${\text{a}}^{\text{2}}{\text{\hspace{0.17em}b}}^{\text{2}}\text{\hspace{0.17em}=\hspace{0.17em}}\frac{\text{1}}{\text{4}}$

${\text{a}}^{\text{2}}\left(\frac{\text{3}}{\text{8}}\right)\text{\hspace{0.17em}=\hspace{0.17em}}\frac{\text{1}}{\text{4}}$

$\Rightarrow {\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}a}}^{\text{2}}\text{\hspace{0.17em}=\hspace{0.17em}}\frac{\text{2}}{\text{3}}$

$\therefore {\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}a}}^{\text{2}}{\text{+b}}^{\text{2}}\text{\hspace{0.17em}=\hspace{0.17em}}\frac{\text{25}}{\text{24}}$