Question

If the normal of the plane makes angles of $\frac{\pi }{4},\frac{\pi }{4}$ and $\frac{\pi }{2}$ with positive x-axis, y-axis and z-axis respectively and the length of the perpendicular line segment from origin to the plane is $\sqrt{2}$, then the equation of the plane is

• A. $x+y+z=\sqrt{2}$
• B. $x+y+z=1$
• C. $x+y=2$
• D. $x=\sqrt{2}$

Answer 1

The correct option is C $x+y=2$

The equation of a plane in normal form is $lx+my+nz=d,$
where $(l,m,n)$ are the direction ratios of the normal to the plane.
and $d$ is the perpendicular distance of the plane from the origin.
Hence, $(l,m,n)=(\cos \alpha ,\cos \beta ,\cos \gamma )$
$\Rightarrow (l,m,n)=(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$
and $d=\sqrt{2}$
Hence, equation of the plane is
$\frac{x}{\sqrt{2}}+\frac{y}{\sqrt{2}}=\sqrt{2}$
$\Rightarrow x+y=2$