The direction ratios of normal to the plane through (1, 0, 0), (0, 1,…

Let the direction ratios of the normal to the plane be proportional to a, b, c. Then, the equation of the plane is

$a(x-1) + b(y-0) + c(z-0) = 0 $   ................(i)

It passes through (0, 1, 0).

$a(-1) + b(1) + c(0) = 0 ⇒ a = b $ ..........(ii)

It is given that the plane (i) makes an angle $\frac{\pi}{4}$ with the plane x + y = 3.

$∴ cos\frac{\pi}{4}=\frac{a×1+b×1+c×0}{\sqrt{a^2+b^2+c^2}\sqrt{1+1}}$

$⇒ \frac{1}{\sqrt{2}}=\frac{a+b}{\sqrt{a^2+b^2+c^2}\sqrt{2}}$

$⇒\sqrt{a^2+b^2+c^2} = a+ b $

$⇒ a^2 + b^2 + c^2 = a^2 + b^2 + 2ab ⇒ c^2 = 2ab$ ...............(iii)

From (ii) and (iii), we have

$a : b : c = a : a : \sqrt{2}a = 1 : 1 : \sqrt{2}$