The combined equation of the lines passing through the origin making an acute angle \(\propto\) with the line \(y=x\) is

The required lines are
\( y=\frac{1+\tan \alpha}{1-\tan \alpha} x \text { and } y=\frac{1-\tan \alpha}{1+\tan \alpha} x \)
\( \Rightarrow(1+\tan \alpha) y=(1+\tan x) x \text { and }(1+\tan \alpha) y=\)\((1-\tan \alpha) x \)
\( \text { joint equation } \)
\( \{1+\tan \alpha) x-(1-\tan \alpha) y\}\{1-\tan \alpha) x-(1+\tan \alpha)\)\( y\}=0 \)
\( \left.\Rightarrow(1-\tan 2 \alpha) x^2-\left\{(1+\tan \alpha)^2\right)+(1-\tan \alpha)^2\right\} x y+\)\(\left(1-\tan ^2 \alpha\right) y^2=0 \)
\( \Rightarrow x^2-2\left(\frac{1+\tan ^2 \alpha}{1-\tan ^2 \alpha}\right)^{x y+y^2=0} \)
\( \Rightarrow x^2-\frac{2}{\cos 2 \alpha} x y+y^2=0 \)
\( \Rightarrow x^2-2 \sec 2 a x y+y^2=0\)