The direction ratios of normal to the plane through the points (0, –1, 0) and (0, 0, 1) and

Question:

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

  1. $2 \sqrt{3}, 1,-1$

  2. $2, \sqrt{2},-\sqrt{2}$

  3. $2,-1,1$

  4. $\sqrt{2}, 1,-1$


Correct Option: , 4

Solution:

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 $\cos \frac{\pi}{4}=\frac{a(0)+b(1)+c(-1)}{\sqrt{2} \sqrt{a^{2}+b^{2}+c^{2}}}$

$\Rightarrow a^{2}=-2 b c$ and $b=-c$

we get $a^{2}=2 c^{2}$

$\Rightarrow \mathrm{a}=\pm \sqrt{2} \mathrm{c}$

$\Rightarrow$ direction ratio $(a, b, c)=(\sqrt{2},-1,1)$ or

$(\sqrt{2}, 1,-1)$

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