The equation of the latus rectum of the ellipse $9{\mathrm{x}}^2+4{\mathrm{y}}^2–18\mathrm{x}–8\mathrm{y}–23=0$ are

$$\begin{gathered} 9({\mathrm{x}}^2-2\mathrm{x}+1-1)+4({\mathrm{y}}^2-2\mathrm{y}+1-1)=23 \\ 9{(x-1)}^2+4{(\mathrm{y}-1)}^2=36 \\ \Rightarrow \frac{{(x-1)}^2}{4}+\frac{{(\mathrm{y}-1)}^2}{9}=1 \\ where centre=(1,1) and a^2=4,b^2=9(a<b) \\ e=\sqrt{\frac{b^2-a^2}{b^2}} \\ \Rightarrow e=\sqrt{\frac{5}{9}}=\frac{\sqrt{5}}{3} \\ Now equation of latusrectum y=k\pm be \\ \Rightarrow y=1\pm 3\left(\frac{\sqrt{5}}{3}\right) \\ \Rightarrow y=1\pm \sqrt{5} \end{gathered}$$