The equation of the normal at p(x, y) to the Ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1\text{ is }\frac{{\mathrm{a}}^2\mathrm{x}}{{\mathrm{x}}_1}-\frac{{\mathrm{b}}^2\mathrm{y}}{{\mathrm{y}}_1}={\mathrm{a}}^2-{\mathrm{b}}^2$

The equation of the tangent to S=0 at P(x₁,y₁) is ${\mathrm{S}}_1=0\Rightarrow \frac{{\mathrm{xx}}_1}{{\mathrm{a}}^2}+\frac{{\mathrm{yy}}_1}{{\mathrm{b}}^2}-1=0 ...\left(1\right)$
The normal at P(x₁, y₁) is perpendicular to (1) and passes through P. Any line perpendicular to (1) is $\frac{{\mathrm{xy}}_1}{{\mathrm{b}}^2}-\frac{{\mathrm{x}}_1\mathrm{y}}{{\mathrm{a}}^2}=\mathrm{k}$
This passes through P(x₁, y₁)
$\Rightarrow \frac{{\mathrm{x}}_1{\mathrm{y}}_1}{{\mathrm{b}}^2}-\frac{{\mathrm{x}}_1{\mathrm{y}}_1}{{\mathrm{a}}^2}=\mathrm{k}$
$\therefore$The equation of the normal is $\frac{{\mathrm{xy}}_1}{{\mathrm{b}}^2}-\frac{{\mathrm{yx}}_1}{{\mathrm{a}}^2}=\frac{{\mathrm{x}}_1{\mathrm{y}}_1}{{\mathrm{b}}^2}-\frac{{\mathrm{x}}_1{\mathrm{y}}_1}{{\mathrm{a}}^2}$
Multiplying both sides by ${\mathrm{a}}^2{\mathrm{b}}^2/{\mathrm{x}}_1{\mathrm{y}}_1,$ 
we get $\frac{{\mathrm{a}}^2\mathrm{x}}{{\mathrm{x}}_1}-\frac{{\mathrm{b}}^2\mathrm{y}}{{\mathrm{y}}_1}={\mathrm{a}}^2-{\mathrm{b}}^2$