If the line of intersection of the planes $\mathrm{ax}+\mathrm{by}=3$ and $\mathrm{ax}+\mathrm{by}+\mathrm{cz}=0$, a > 0 makes an angle 30° with the plane $\mathrm{y}-\mathrm{z}+2=0,$ then the direction cosines of the line are :

$\begin{array}{l}\overrightarrow{\mathrm{n}}=\left|\begin{array}{ccc}\hat{\mathrm{i}}& \hat{\mathrm{j}}& \hat{\mathrm{k}}\\ \mathrm{a}& \mathrm{b}& 0\\ \mathrm{a}& \mathrm{b}& \mathrm{c}\end{array}\right|\\ =\mathrm{bc}\hat{\mathrm{i}}-\mathrm{ac}\hat{\mathrm{j}}\end{array}$

Direction ratios of line are b, –a, 0

Direction ratios of normal of the plane are 0, 1, – 1

$\begin{array}{l}\cos {60}^{\circ }=\left|\frac{-\mathrm{a}}{\left|\sqrt{2}\right|\sqrt{{\mathrm{b}}^2+{\mathrm{a}}^2}}\right|=\frac{1}{2}\\ \Rightarrow \left|\frac{\mathrm{a}}{\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}}\right|=\frac{1}{\sqrt{2}}\\ \Rightarrow \mathrm{b}=\pm \mathrm{a}\end{array}$

So, D.R.’s can be $\left(\pm \mathrm{a}, -\mathrm{a}, 0\right)$

$\therefore$ D.C.’s can be $\pm \left(\frac{\pm 1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}, 0\right)$