Question

If the direction cosines of the line joining the origin and a point at unit distance from the origin are $\frac{1}{\sqrt{3}}$, $-\frac{1}{2}$, $\lambda$ then value of $\lambda$ is?

• A. $\frac{1}{2}\sqrt{\frac{5}{3}}$
• B. $\frac{2}{\sqrt{3}}$
• C. $\frac{-1}{2}\sqrt{\frac{5}{3}}$
• D. none of these

Answer 1

The correct option is A $\frac{1}{2}\sqrt{\frac{5}{3}}$

Direction cosines of the line are given as $\frac{1}{\sqrt{3}},\frac{-1}{2},\lambda$
We know that
${\cos }^{2}a+{\cos }^{2}b+{\cos }^{2}c=1$
i.e ${\frac{1}{\sqrt{3}}}^2+{\frac{-1}{2}}^2+{\lambda }^2=1$
$\frac{1}{3}+\frac{1}{4}+{\lambda }^2=1$
$\frac{7}{12}+{\lambda }^2=1$
${\lambda }^2=\frac{5}{12}$
Therefore, $\lambda =\frac{\sqrt{5}}{2\sqrt{3}}$

Hence, option A is correct.