A conic has latus rectum length 1, focus at (2, 3) and the corresponding directrix is x + y – 3 = 0. Then the conic is 
 

$$\begin{gathered} \mathrm{Le}ngth of \mathrm{L}.\mathrm{R}=\frac{2{\mathrm{b}}^2}{\mathrm{a}}=1 \\ \mathrm{a}=2{\mathrm{b}}^2=2{\mathrm{a}}^2(1-{\mathrm{e}}^2) \\ 1=2\mathrm{a}(1-{\mathrm{e}}^2) \\ \frac{\mathrm{a}}{\mathrm{e}}-\mathrm{ae}=\left|\frac{2+3-3}{\sqrt{1+1}}\right|=\sqrt{2} \\ \frac{\mathrm{a}(1-{\mathrm{e}}^2)}{\mathrm{e}}=\sqrt{2} \\ \frac{1}{2\mathrm{e}}=\sqrt{2}\Rightarrow \mathrm{e}=\frac{1}{2\sqrt{2}}<1 \\ where 0<e<1 \\ \therefore \mathrm{conic} \mathrm{is} an \mathrm{ellipse} \end{gathered}$$