The direction ratios of normal to the plane through the points (0,-1,0) and (0,0,1) and making an angle…
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;
2,-1,1
$2, \sqrt{2}-\sqrt{2}$
$\sqrt{2}, 1,-1$
$2 \sqrt{3}, 1,-1$
option 1 and 2
option 2 and 3
option 3 and 4
all the options
Solution
Let the d.r's of the normal be $\langle a, b, c\rangle$ Equation of the plane is $a(x-0)+b(y+1)+c(z-0)=0$
$\because$ It passes through (0,0,1)
$\therefore \quad b+c=0$
Also $\frac{0 \cdot a+b-c}{\sqrt{a^{2}+b^{2}+c^{2} \cdot \sqrt{2}}}=\cos \frac{\pi}{4}=\frac{1}{\sqrt{2}}$
$\Rightarrow \quad b-c=\sqrt{a^{2}+b^{2}+c^{2}}$
And $b+c=0$
$\Rightarrow \quad b=\pm \frac{1}{\sqrt{2}} a$
$\therefore \quad$ The d.r's are $\sqrt{2}, 1,-1$ or $2, \sqrt{2},-\sqrt{2}$