Question

A plane passing through the points $(0,-1,0)$ and $(0,0,1)$ and making an angle $\frac{\pi }{4}$ with the plane $y-z+5=0$, also passes through the point:

• A. ($\sqrt{2},-1,4)$
• B. $(-\sqrt{2},-1,-4)$
• C. ($-\sqrt{2},1,-4)$
• D. ($\sqrt{2},1,4)$

Answer 1

The correct option is A ($\sqrt{2},-1,4)$

Let $ax+by+cz=1$ be the equation of the plane.
As it passes through $(0,-1,0)$ and $(0,0,1)$,
$0-b+0=1\Rightarrow b=-1and0+0+c=1\Rightarrow c=1$
We know,
$\cos \theta =\left|\frac{\overrightarrow{a}\cdot \overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}\right|\cos \frac{\pi }{4}=\frac{|0-1-1|}{\sqrt{(a^2+1+1)(0+1+1)}}\Rightarrow \frac{1}{\sqrt{2}}=\frac{2}{\sqrt{a^2+2}\sqrt{2}}\Rightarrow a^2+2=4\Rightarrow a=\pm \sqrt{2}$
Therefore, equation of the plane is
$\pm \sqrt{2}x-y+z=1$
By substituting all the points, we can see that only option (a) satisfies the equation of plane for -ve sign.