Question

The direction ratios of normal to the plane through the points $(0,-1,0)$ and $(0,0,1)$ and making an angle $\frac{\pi }{4}$ with the plane $y-z+5=0$ are :

• A. $2\sqrt{3},1-1$
• B. $2,\sqrt{2},-\sqrt{2}$
• C. $2,-1,1$
• D. $\sqrt{2},1,-1$

Answer 1

Answer: (B), (D)

Let the equation of plane be $a(x-0)+b(y+1)+c(z-0)=0$
It passes through $(0,0,1)$ then $b+c=0$ .....(1)
Now $\cos \frac{\pi }{4}=\frac{a\left(0\right)+b\left(1\right)+c\left(-1\right)}{\sqrt{2}\sqrt{a^2+b^2+c^2}}$
$\Rightarrow a^2=-2bc$ and $b=-c$
we get $a^2=2c^2$
$\Rightarrow a=\pm \sqrt{2}c$
$\Rightarrow$ direction ratio (a, b, c) = $(\sqrt{2},-1,1)$ or $(\sqrt{2},1,-1)$