Question

If the angle between the line $2(x+1)=y=z+4$ and the plane $2x-y+\sqrt{\lambda }z+4=0$ is $\frac{\pi }{6}$, then the value of $\lambda$ is :

• A. $\frac{135}{7}$
• B. $\frac{45}{11}$
• C. $\frac{45}{7}$
• D. $\frac{135}{11}$

Answer 1

The correct option is C $\frac{45}{7}$

Given equation of line

$\frac{x-(-1)}{1/2}=\frac{y-0}{1}=\frac{z-(-4)}{1}$

Equation of plane is

$2x-y+\sqrt{\lambda }z+4=0$ .

$\sin \frac{\pi }{6}=\frac{\frac{1}{2}\times 2+1(-1)+1\sqrt{\lambda }}{\sqrt{{\left(\frac{1}{2}\right)}^2+{\left(1\right)}^2+{\left(1\right)}^2}\sqrt{{\left(2\right)}^2+{(-1)}^2+{\left(\sqrt{\lambda }\right)}^2}}$

$\Rightarrow 2\sqrt{\lambda }=\frac{3}{2}\sqrt{5+\lambda }$

$\Rightarrow \lambda =\frac{45}{7}$