Question

The equation of plane parallel to the plane $2x-y+2z=5$ and is at unit distance from origin, is(are)

• A. $2x-y+2z+3=0$
• B. $2x-y+2z+5=0$
• C. $2x-y+2z-3=0$
• D. $2x-y+2z-2=0$

Answer 1

Answer: (A), (C)

$2x-y+2z-3=0$

As required plane is parallel to $2x-y+2z=5$
$i.e.$ both plane have same normal $\overrightarrow{n}=2\hat{i}-\hat{j}+2\hat{k}$
unit vector along normal is: $\hat{n}=\frac{2\hat{i}-\hat{j}+2\hat{k}}{3}$
So, equation of plane is given by : $2x-y+2z+d=0$
converting it into normal form,
$\Rightarrow \frac{2x}{3}-\frac{y}{3}+\frac{2z}{3}+\frac{d}{3}=0\Rightarrow \frac{2x}{3}-\frac{y}{3}+\frac{2z}{3}=-\frac{d}{3}$
$\Rightarrow \left|\frac{d}{3}\right|=1\Rightarrow d=\pm 3$
So, equation of plane is : $2x-y+2z\pm 3=0$