Let $\mathrm{x} \mathrm{tan\alpha }+\mathrm{y} \sin \mathrm{\alpha }=\mathrm{\alpha }$ and $\mathrm{x} \mathrm{\alpha } \mathrm{cosec} \mathrm{\alpha }+\mathrm{y} \cos \mathrm{\alpha }=1$ be two variable lines, $\alpha$ being parameter. Let P be the point of intersection of lines in the limiting position when $\alpha \to 0$, the point P lies on the line

 $$\begin{gathered} point of intersection \\ \mathrm{x}=\frac{\mathrm{\alpha cos\alpha }-\mathrm{sin\alpha }}{\mathrm{sin\alpha }-\mathrm{\alpha }},y=\frac{\alpha -x\tan \alpha }{\sin \alpha } \\ now \underset{\mathrm{\alpha }\to 0}{\lim }\mathrm{x}=\underset{\mathrm{\alpha }\to 0}{\lim }\frac{\mathrm{\alpha cos\alpha }-\mathrm{sin\alpha }}{\mathrm{sin\alpha }-\mathrm{\alpha }}=\underset{\mathrm{\alpha }\to 0}{\lim }\frac{\mathrm{cos\alpha }-\alpha \mathrm{sin\alpha }-\mathrm{cos\alpha }}{\mathrm{cos\alpha }-1} \\ =\underset{\mathrm{\alpha }\to 0}{\lim }\frac{{\displaystyle \frac{\mathrm{sin\alpha }}{\alpha }} }{{\displaystyle \frac{1-\mathrm{cos\alpha }}{{\alpha }^2}}} =\frac{1}{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}= 2 \end{gathered}$$

$\underset{\alpha \to 0}{lim} y=\underset{\alpha \to \alpha }{lim} \frac{\alpha -x\tan \alpha }{\sin \alpha }=\underset{\alpha \to 0}{lim} \left(\frac{\alpha }{\sin \alpha }-\frac{x}{\cos \alpha }\right)=1-2=-1$