Question

If the direction ratios of a line are $1+\lambda ,1-\lambda ,2,$ and it makes an angle of ${60}^o$ with the y-axis then $\lambda$ is

• A. $1+\sqrt{3}$
• B. $2+\sqrt{5}$
• C. $1-\sqrt{3}$
• D. $2-\sqrt{5}$

Answer 1

Answer: (B), (D)

The correct options are
B $2+\sqrt{5}$
D $2-\sqrt{5}$

If $a,b,c$ are direction ratios and $\theta$ is the angle between y axis and line.

Then $\cos \theta =\frac{b}{\sqrt{a^2+b^2+c^2}}$.

Hence by the given informations we have

$\frac{1}{4}={\cos }^2\left(60\right)=\frac{(1-\lambda {)}^2}{(1+\lambda {)}^2+(1-\lambda {)}^2+4}$

$\Rightarrow {\lambda }^2-4\lambda -1$

$\Rightarrow \lambda =2\pm \sqrt{5}$