Question

The direction ratios of a normal to the plane through $(1, 0, 0), (0, 1, 0)$ which makes angles of $\frac{\pi}{4}$ 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

Answer: (B)

$<1,1,\sqrt 2>$

Answer 2

Any plane through $(1, 0, 0)$ is $a(x - 1) + b (y - 0) + c (z - 0)$ = 0 ....(1) It contains (0, 1, 0) if - a + b = 0 ....(2) Also (1) makes and angle of $\frac{\pi}{4}$ with the plane $x + y = 3$ $\therefore$ cos$\frac{\pi}{4} = \frac{a + b}{\sqrt{a^2 + b^2 + c^2} \sqrt{1+1}}$ $\Rightarrow$ $(a + b)^2 = a^2 + b^2 + c^2$ $\Rightarrow$ $2ab = c^2$ $\Rightarrow$ $2a^2 = c^2$ [$\because \, a = b$ from (2)] $\Rightarrow$ $c = \sqrt{2} a$ $\therefore \, a : b : c = a : a : \sqrt{2} a = 1 : 1 : \sqrt{2}$ Hence reqd. direction ratios are < 1, 1, $\sqrt{2}$ >.