If the angel [ ] between the line [x + 1 1 = y - 1 2 = z - 2 2] and the plane [2x - y + ,,…

Question

If the angel $$\theta $$ between the line $${{x + 1} \over 1} = {{y - 1} \over 2} = {{z - 2} \over 2}$$ and

the plane $$2x - y + \sqrt \lambda  \,\,z + 4 = 0$$ is such that $$\sin \,\,\theta  = {1 \over 3}$$ then value of $$\lambda $$ is :

Accepted Answer

If $$\theta $$ is the angle between line and plane then $$\left( {{\pi  \over 2} - 0} \right)$$

is the angle between line and normal to plane given by

$$\cos \left( {{\pi  \over 2} - 0} \right) = {{\left( {\widehat i + 2\widehat j + 2\widehat k} \right).\left( {2\widehat i - \widehat j + \sqrt \lambda  \widehat k} \right)} \over {3\sqrt {4 + 1 + \lambda } }}$$

$$\cos \left( {{\pi  \over 2} - \theta } \right) = {{2 - 2 + 2\sqrt \lambda  } \over {3 \times \sqrt 5  + \lambda }}$$

$$ \Rightarrow \sin \theta  = {{2\sqrt \lambda  } \over {3\sqrt 5  + \lambda }} = {1 \over 3}$$

$$ \Rightarrow 4\lambda  = 5 + \lambda  \Rightarrow \lambda  = {5 \over 3}$$

Solutions · 1