Find the centre, length of major axis, minor axis, eccentricity, foci, length of latusrectum and equation of directrices of the ellipses. 

(i) $4{\mathrm{x}}^2+{\mathrm{y}}^2-8\mathrm{x}+2\mathrm{y}+1=0$

(ii)  $9{\mathrm{x}}^2+16{\mathrm{y}}^2-36\mathrm{x}+32\mathrm{y}-92=0$

$\text{ (iii) }9{\mathrm{x}}^2+16{\mathrm{y}}^2=144$

i) $4{\mathrm{x}}^2+{\mathrm{y}}^2-8\mathrm{x}+2\mathrm{y}+1=0$
$\begin{array}{l}4{\mathrm{x}}^2-8\mathrm{x}+{\mathrm{y}}^2+2\mathrm{y}+1=0\\ 4\left({\mathrm{x}}^2-2\mathrm{x}\right)+(\mathrm{y}+1{)}^2=0\\ 4\left({\mathrm{x}}^2-2\mathrm{x}+1\right)+(\mathrm{y}+1{)}^2=4\\ 4(\mathrm{x}-1{)}^2+(\mathrm{y}+1{)}^2=4\\ \frac{(\mathrm{x}-1{)}^2}{1}+\frac{(\mathrm{y}+1{)}^2}{4}=1\end{array}$
Compare with $\frac{(\mathrm{x}-\mathrm{h}{)}^2}{{\mathrm{a}}^2}+\frac{(\mathrm{y}-\mathrm{k}{)}^2}{{\mathrm{b}}^2}=1$
$\mathrm{a}=1,\mathrm{b}=2,\mathrm{a}<\mathrm{b}$
$\text{ Centre }\mathrm{c}(\mathrm{h},\mathrm{k})=(1,-1)$
Length of major axis 2b = 2(2) = 4
$\text{ Length of minor axis }2\mathrm{a}=2\left(1\right)=2$
$\text{ Eccentricity }\mathrm{e}=\sqrt{\frac{{\mathrm{b}}^2-{\mathrm{a}}^2}{{\mathrm{b}}^2}}=\sqrt{\frac{4-1}{4}}=\frac{\sqrt{3}}{2}$
Foci are $(\mathrm{h},\mathrm{k}\pm \mathrm{be})=\left(1,-1\pm 2\frac{\sqrt{3}}{2}\right)=(1,-1\pm \sqrt{3})$
Length of latus rectum $=\frac{2{\mathrm{a}}^2}{\mathrm{b}}=\frac{2.1}{2}=1$
Equation of directrix $\mathrm{y}=\mathrm{k}\pm \frac{\mathrm{b}}{\mathrm{e}}$
$\begin{array}{l}\mathrm{y}=-1\pm \frac{2}{\sqrt{3}/2}\\ \mathrm{y}=-1\pm \frac{4}{\sqrt{3}}\\ \sqrt{3}\mathrm{y}+\sqrt{3}\pm 4=0\end{array}$
ii) $9{\mathrm{x}}^2+16{\mathrm{y}}^2-36\mathrm{x}+32\mathrm{y}-92=0$
$\begin{array}{l}9{\mathrm{x}}^2-36\mathrm{x}+16{\mathrm{y}}^2+32\mathrm{y}-92=0\\ 9\left({\mathrm{x}}^2-4\mathrm{x}\right)+16\left({\mathrm{y}}^2+2\mathrm{y}\right)=92\\ 9\left({\mathrm{x}}^2-4\mathrm{x}+4\right)+16\left({\mathrm{y}}^2+2\mathrm{y}+1\right)=92+36+16\\ 9(\mathrm{x}-2{)}^2+16(\mathrm{y}+1{)}^2=144\end{array}$
$\frac{(\mathrm{x}-2{)}^2}{16}+\frac{(\mathrm{y}+1{)}^2}{9}=1$
Compare $\frac{(\mathrm{x}-\mathrm{h}{)}^2}{{\mathrm{a}}^2}+\frac{(\mathrm{y}-\mathrm{k}{)}^2}{{\mathrm{b}}^2}=1$
$\mathrm{a}=4,\mathrm{b}=3,\mathrm{a}>\mathrm{b}$
Centre (h,k) = (2,-1)
$\text{ Length of major axis }=2\mathrm{a}=2\left(4\right)=8$
$\text{ Length of minor axis }=2\mathrm{b}=2\left(3\right)=6$
$\text{ Eccentricity }\mathrm{e}=\sqrt{\frac{{\mathrm{a}}^2-{\mathrm{b}}^2}{{\mathrm{a}}^2}}=\sqrt{\frac{16-9}{16}}=\frac{\sqrt{7}}{4}$
$\begin{array}{l}\mathrm{Foci} \mathrm{are} (\mathrm{h}\pm \mathrm{ae},\mathrm{k})=\left(2\pm 4\cdot \frac{\sqrt{7}}{4},-1\right)\\ =(2\pm \sqrt{7},-1)\end{array}$
Length of latus rectum $=\frac{2{\mathrm{b}}^2}{\mathrm{a}}=\frac{2\left(9\right)}{4}=\frac{9}{2}$
Equation of directrices are $\mathrm{x}=\mathrm{h}\pm \frac{\mathrm{a}}{\mathrm{e}}$
$\begin{array}{l}\mathrm{x}=2\pm \frac{4}{\sqrt{7}/4}\Rightarrow \mathrm{x}=2\pm \frac{16}{\sqrt{7}}\\ \sqrt{7}\mathrm{x}=2\sqrt{7}\pm 16\end{array}$
$\text{ iii) }9{\mathrm{x}}^2+16{\mathrm{y}}^2=144$
$$\begin{gathered} \frac{{\mathrm{x}}^2}{16}+\frac{{\mathrm{y}}^2}{9}=1\mathrm{a}=4,\mathrm{b}=3,\mathrm{a}>\mathrm{b} \\ \text{ Centre }\mathrm{c}(\mathrm{h},\mathrm{k})=(0,0) \\ \text{ Length of major axis }=2\mathrm{a}=2\left(4\right)=8 \\ \text{ Length of minor axis }=2\mathrm{b}=2\left(3\right)=6 \\ \text{ Eccentricity }\mathrm{e}=\sqrt{\frac{{\mathrm{a}}^2-{\mathrm{b}}^2}{{\mathrm{a}}^2}}=\sqrt{\frac{16-9}{16}}=\frac{\sqrt{7}}{4} \end{gathered}$$
$$\begin{gathered} \text{ Foci are }(\pm \mathrm{ae},0)=\left(\pm 4\cdot \frac{\sqrt{7}}{4},0\right)=(\pm \sqrt{7},0) \\ \text{ Length of latus rectum }=\frac{2{\mathrm{b}}^2}{\mathrm{a}}=\frac{2\left(9\right)}{4}=\frac{9}{2} \\ \text{ equation of directrices are }\mathrm{x}=\pm \frac{\mathrm{a}}{\mathrm{e}} \\ \mathrm{x}=\pm \frac{4}{\frac{\sqrt{7}}{4}}\Rightarrow \mathrm{x}=\pm \frac{16}{\sqrt{7}}\Rightarrow \sqrt{7}\mathrm{x}\pm 16=0 \end{gathered}$$