The equations of latus recta of the ellipse $9x^2+4y^2-18x-8y-23=0\text{ are }$

$\begin{array}{l}9x^2+4y^2-18x-8y-23=0\Rightarrow 9\left(x^2-2x+1\right)+4\left(y^2-2y+1\right)=36\\ \Rightarrow \frac{(x-1{)}^2}{4}+\frac{(y-1{)}^2}{9}=1\Rightarrow \text{ Centre }=(1,1),a=2,b=3\end{array}$

Eccentricity $e=\frac{\sqrt{9-4}}{3}=\frac{\sqrt{5}}{3},\text{ Foci are }(1,1\pm \sqrt{5})$ 

Equations of the required latusrecta are y=k$\pm$be