Question

If the $D.R^{\prime }s$ of a line are $(1+\lambda ,1-\lambda ,2)$ and it makes an angle of ${60}^{\circ }$ with $Y$ axis, then the value of $\lambda$ can be

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

Answer 1

Answer: (A), (D)

$2-\sqrt{5}$

we know,
$D.C^{\prime }s$ of $y-$axis is $(0,1,0)$ and
$D.R^{\prime }s$ of given line $=(1+\lambda ,1-\lambda ,2)$
So, angle between two lines,
$\cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$
$\frac{1}{2}=\frac{1-\lambda }{\sqrt{(1+\lambda {)}^2+(1-\lambda {)}^2+4}\times \sqrt{1}}$
squaring both sides and simplifying we get,
$\Rightarrow 2{\lambda }^2+6=4({\lambda }^2-2\lambda +1)\Rightarrow {\lambda }^2-4\lambda -1=0$
$\Rightarrow \lambda =2\pm \sqrt{5}$