Explanation

(b)
Any plane through (1, 0, 0) is $a(x - 1) + by + cz = 0\ \ \$(i)
It passes through (0, 1, 0)
$\therefore\ a(0 - 1) + b(1) + c(0) = 0\ \Rightarrow - a + b = 0$ (ii)
(i) makes an angle of $\frac{\pi}{4}$ with $x + y = 3$, therefore
$\cos{\frac{\pi}{4} = \frac{a(1) + b(1) + c(0)}{\sqrt{a^{2} + b^{2} + c^{2}}\sqrt{1 + 1 + 0}}}$
$\Rightarrow \ \frac{1}{\sqrt{2}} = \frac{a + b}{\sqrt{2}\sqrt{a^{2} + b^{2} + c^{2}}}$
$\Rightarrow a + b = \sqrt{a^{2} + b^{2} + c^{2}}$
Squaring, we get
$a^{2} + b^{2} + 2ab = a^{2} + b^{2} + c^{2}$
$\Rightarrow 2ab = c^{2}\ \Rightarrow 2a^{2} = c^{2}$
$\Rightarrow c = \sqrt{2}\ a$ (using (ii))
Hence, $a:b:c = a:a:\sqrt{2}a$
$= 1:1:\sqrt{2}$