Question

The direction ratios of a normal to the plane passing through (1, 0, 0), (0, 1, 0) and making an angle $\frac{\pi }{4}$ with the plane x + y = 3 are :

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

Answer 1

The correct option is B $(1,1\sqrt{2})$

Any plane through (1, 0, 0), (0, 1, 0) can be taken as
$\frac{x}{1}+\frac{y}{1}+\frac{z}{c}=1$

D.R.’s of normal to this plane are 1, 1, $\frac{1}{c}$
This plane is inclined at an angle $\frac{\pi }{4}$ with the plane x + y = 3
$\therefore cos\left(\frac{\pi }{4}\right)=\frac{1+1+\frac{0.1}{c}}{\sqrt{1^2+1^2+\frac{1}{c^2}}\sqrt{1+1}}\Rightarrow \frac{4c^1}{(2c^2+1{)}^2}=\frac{1}{2}$
$\Rightarrow \frac{1}{2}$ i.e., c $=\pm \frac{1}{\sqrt{2}}$
$\therefore$ D.R.’s of normal of the required plane are $(1,1,\sqrt{2})$ or $(1,1-\sqrt{2})$