Question

If the normal of the plane makes an angles $\frac{\pi }{4},\frac{\pi }{4}$ 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 $x+y=2$

Normal of the plane makes an angle of $\frac{\pi }{4},\frac{\pi }{4},\frac{\pi }{2}$ with X-axix,Y-axis, and Z-axis respectively,

Direction ratio of normal to the plane is$(\cos \alpha ,\cos \beta ,\cos \gamma )$

Direction ratio of normal to the plane is

$(\cos \frac{\pi }{4},\cos \frac{\pi }{4},\cos \frac{\pi }{2})$
$(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$

Equation of the plane is given by

$x\cos \alpha +y\cos \beta +z\cos \gamma =\perp distance$
$\therefore \frac{x}{\sqrt{2}}+\frac{y}{\sqrt{2}}=\sqrt{2}$
$x+y=2$.