Example 2. Derive the equations of the tangent and normal to the ellipse

$$
\frac{x^{2}}{18}+\frac{y^{2}}{8}=1
$$

at the point $M_{0}(3,2)$.
Solution. We find $\frac{x^{2}}{18}+\frac{y^{2}}{8}-1=0$ :

$$
\frac{2 x}{18}+\frac{2 y y^{\prime}}{8}=0,
$$

from which

$$
y^{\prime}(x)=-\frac{4 x}{9 y}, y^{\prime}\left(x_{0}\right)=y^{\prime}(3)=-\frac{4 \cdot 3}{2 \cdot 9}=-\frac{2}{3}
$$

Substituting the values $x_{0}=3, y_{0}=2, y^{\prime}\left(x_{0}\right)=-\frac{2}{3}$ into formulas (3.1) and $(3.2)$, we get

$$
\begin{gathered}
y-2=-\frac{2}{3}(x-3) \text { or } 2 x+3 y-12=0-\text { equation of the tangent; } \\
y-2=\frac{3}{2}(x-3) \text { or } 3 x-2 y-5=0-\text { equation of the normal. }
\end{gathered}
$$