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

Answer 1

The correct option is A $\sqrt{2},1,-1$

Direction of ratios of line through $A(0,-1,0)$ and $B(0,0,1)$ is $(0,1,1)$
Let DR's of normal to plane are $(a,b,c)$
$\therefore b+c=0$
$\cos \frac{\pi }{4}=\frac{a\left(0\right)+b\left(1\right)+c(-1)}{\sqrt{a^2+b^2+c^2}\sqrt{(1{)}^2+(-1{)}^2}}$
$\Rightarrow \frac{1}{\sqrt{2}}=\frac{b-c}{\sqrt{a^2+b^2+c^2}\times \sqrt{2}}$
$\Rightarrow a^2+b^2+c^2=b^2+c^2-2bc$
$\Rightarrow a^2=-2bc$
$\Rightarrow a^2=-2b(-b)$
$\Rightarrow a^2=2b^2$
$\Rightarrow a=\sqrt{2}b$
$\therefore$ DR's of normal to plane are $(\sqrt{2},-1,1)$ or $(\sqrt{2},1,-1)$