Question
A plane passing through the points $$\left( {0,\, - 1,\,0} \right)$$ and $$\left( {0,\,0,\,1} \right)$$ and making an angle $$\frac{\pi }{4}$$ with the plane $$y-z+5=0,$$ also passes through the point :
A.
$$\left( { - \sqrt 2 ,\,1,\, - 4} \right)$$
B.
$$\left( { \sqrt 2 ,\, - 1,\,4} \right)$$
C.
$$\left( { - \sqrt 2 ,\, - 1,\, - 4} \right)$$
D.
$$\left( {\sqrt 2 ,\,1,\,4} \right)$$
Answer :
$$\left( {\sqrt 2 ,\,1,\,4} \right)$$
Solution :
Let the required plane passing through the points $$\left( {0,\, - 1,\,0} \right)$$ and $$\left( {0,\,0,\,1} \right)$$ be $$\frac{x}{\lambda } + \frac{y}{{ - 1}} + \frac{z}{1} = 1$$ and the given plane is $$y-z+5=0$$
$$\eqalign{
& \therefore \,\,\,\cos \frac{\pi }{4} = \frac{{ - 1 - 1}}{{\sqrt {\left( {\frac{1}{{{\lambda ^2}}} + 1 + 1} \right)} \sqrt 2 }} \cr
& \Rightarrow {\lambda ^2} = \frac{1}{2} \cr
& \Rightarrow \frac{1}{\lambda } = \pm \sqrt 2 \cr} $$
Then, the equation of plane is $$ \pm \sqrt 2 x - y + z = 1$$
Then the point $$\left( {\sqrt 2 ,\,1,\,4} \right)$$ satisfies the equation of plane $$ - \sqrt 2 x - y + z = 1$$