The equation of the chord of the ellipse $4x^2+9y^2=36$ having $(3,2)$ as midpoint is

$4x^2+9y^2=36\Rightarrow \frac{x^2}{9}+\frac{y^2}{4}=1\Rightarrow a^2=9,b^2=4$

Midpoint $(3,2)=(x_1,y_1)$ 

Equation of chord $S_1=S_{11}$

$\frac{x{x}_1}{a^2}+\frac{y{y}_1}{b^2}=\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}$

$\frac{x.3}{9}+\frac{y.2}{4}=\frac{9}{9}+\frac{4}{4},\frac{3x}{9}+\frac{2y}{4}=2$

$\frac{x}{3}+\frac{y}{2}=2\Rightarrow 2x+3y=12$