Question

Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:
(i) x² + 2y² − 2x + 12y + 10 = 0
(ii) x² + 4y² − 4x + 24y + 31 = 0
(iii) 4x² + y² − 8x + 2y + 1 = 0
(iv) 3x² + 4y² − 12x − 8y + 4 = 0
(v) 4x² + 16y² − 24x − 32y − 12 = 0
(vi) x² + 4y² − 2x = 0

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)x^2+2y^2-2x+12y+10=0 \\ \Rightarrow \left(x^2-2x\right)+2\left(y^2+6y\right)=-10 \\ \Rightarrow \left(x^2-2x+1\right)+2\left(y^2+6y+9\right)=-10+18+1 \\ \Rightarrow {\left(x-1\right)}^2+2{\left(y+3\right)}^2=9 \\ \Rightarrow \frac{{\left(x-1\right)}^2}{9}+\frac{{\left(y+3\right)}^2}{{\displaystyle \frac{9}{2}}}=9 \\ \mathrm{Here},x_1=1and{y}_1=-3 \\ \mathrm{Also},a=3andb=\frac{3}{\sqrt{2}} \\ \text{Centre=}\left(1,-3\right) \\ \text{Major axis=2}\mathit{\text{a}} \\ \Rightarrow 2\times 3=6 \\ \text{Minor axis=2}\mathit{\text{b}} \\ \Rightarrow 2\times \frac{3}{\sqrt{2}}=3\sqrt{2} \\ e=\sqrt{1-\frac{b^2}{a^2}} \\ \Rightarrow e=\sqrt{1-\frac{{\displaystyle \frac{9}{2}}}{9}} \\ \Rightarrow e=\frac{1}{\sqrt{2}} \\ \mathrm{Foci}=\left(x_1\pm ae,y_1\right) \\ =\left(1\pm \frac{3}{\sqrt{2}},-3\right) \end{gathered}$$
$$\begin{gathered} \left(\mathrm{ii}\right)x^2+4y^2-4x+24y+31=0 \\ \Rightarrow \left(x^2-4x\right)+4\left(y^2+6y\right)=-31 \\ \Rightarrow \left(x^2-4x+4\right)+4\left(y^2+6y+9\right)=-31+36+4 \\ \Rightarrow {\left(x-2\right)}^2+4{\left(y+3\right)}^2=9 \\ \Rightarrow \frac{{\left(x-2\right)}^2}{9}+\frac{{\left(y+3\right)}^2}{{\displaystyle \frac{9}{4}}}=9 \\ \mathrm{Here},x_1=2\mathrm{and}{y}_1=-3 \\ Also,a=3andb=\frac{3}{2} \\ \text{Centre=}\left(x_1,y_1\right)\text{=}\left(2,-3\right) \\ \text{Major axis=2}\mathit{\text{a}} \\ =2\times 3 \\ =6 \\ \text{Minor axis=2}\mathit{\text{b}} \\ =2\times \frac{3}{2} \\ =3 \\ e=\sqrt{1-\frac{b^2}{a^2}} \\ \Rightarrow e=\sqrt{1-\frac{{\displaystyle \frac{9}{4}}}{9}} \\ \Rightarrow e=\frac{\sqrt{3}}{2} \\ \mathrm{Foci}=\left(x_1\pm ae,y_1\right) \\ =\left(2\pm \frac{3\sqrt{3}}{2},-3\right) \end{gathered}$$
$$\begin{gathered} \left(\mathrm{iii}\right)4x^2+y^2-8x+2y+1=0 \\ \Rightarrow 4\left(x^2-2x\right)+\left(y^2+2y\right)=-1 \\ \Rightarrow 4\left(x^2-2x+1\right)+\left(y^2+2y+1\right)=-1+4+1 \\ \Rightarrow 4{\left(x-1\right)}^2+{\left(y+1\right)}^2=4 \\ \Rightarrow \frac{{\left(x-1\right)}^2}{1}+\frac{{\left(y+1\right)}^2}{{\displaystyle 4}}=1 \\ \mathrm{Here},x_1=1and{y}_1=-1 \\ Also,a=1andb=2 \\ \text{Centre=}\left(x_1,y_1\right)\text{=}\left(1,-1\right) \\ \text{Major axis=2}\mathit{\text{b}} \\ =2\times 2 \\ =4 \\ \text{Minor axis=2}\mathit{\text{a}} \\ =2\times 1 \\ =2 \\ e=\sqrt{1-\frac{a^2}{b^2}} \\ \Rightarrow e=\sqrt{1-\frac{{\displaystyle 1}}{4}} \\ \Rightarrow e=\frac{\sqrt{3}}{2} \\ \mathrm{Foci}=\left(x_1,y_1\pm be\right) \\ =\left(1,-1\pm \sqrt{3}\right) \end{gathered}$$
$$\begin{gathered} \left(\mathrm{iv}\right)3x^2+4y^2-12x-8y+4=0 \\ \Rightarrow 3\left(x^2-4x\right)+4\left(y^2-2y\right)=-4 \\ \Rightarrow 3\left(x^2-4x+4\right)+4\left(y^2-2y+1\right)=-4+12+4 \\ \Rightarrow 3{\left(x-2\right)}^2+4{\left(y-1\right)}^2=12 \\ \Rightarrow \frac{{\left(x-2\right)}^2}{4}+\frac{{\left(y-1\right)}^2}{{\displaystyle 3}}=1 \\ \mathrm{Here},x_1=2,y_1=1 \\ Also,a=2andb=\sqrt{3} \\ \text{Centre=}\left(x_1,y_1\right)\text{=}\left(2,1\right) \\ \text{Major axis=2}\mathit{\text{a}} \\ =2\times 2 \\ =4 \\ \text{Minor axis=2}\mathit{\text{b}} \\ =2\times \sqrt{3} \\ =2\sqrt{3} \\ e=\sqrt{1-\frac{b^2}{a^2}} \\ \Rightarrow e=\sqrt{1-\frac{{\displaystyle 3}}{4}} \\ \Rightarrow e=\frac{1}{2} \\ \mathrm{Foci}=\left(x_1\pm ae,y_1\right) \\ =\left(2\pm 1,1\right) \end{gathered}$$
$$\begin{gathered} \left(\mathrm{v}\right)4x^2+16y^2-24x-32y-12=0 \\ \Rightarrow 4\left(x^2-6x\right)+16\left(y^2-2y\right)=12 \\ \Rightarrow 4\left(x^2-6x+9\right)+16\left(y^2-2y+1\right)=12+36+16 \\ \Rightarrow 4{\left(x-3\right)}^2+16{\left(y-1\right)}^2=64 \\ \Rightarrow \frac{{\left(x-3\right)}^2}{16}+\frac{{\left(y-1\right)}^2}{{\displaystyle 4}}=9 \\ \text{Centre=}\left(3,1\right) \\ \mathrm{M}\text{ajor axis=2}\mathit{\text{a}} \\ =2\times 4 \\ =8 \\ \text{Minor axis=2}\mathit{\text{b}} \\ =2\times 2 \\ =4 \\ e=\sqrt{1-\frac{b^2}{a^2}} \\ \Rightarrow e=\sqrt{1-\frac{{\displaystyle 4}}{16}} \\ \Rightarrow e=\frac{\sqrt{3}}{2} \\ \mathrm{Foci}=\left(x\pm ae,y\right) \\ =\left(3\pm 2\sqrt{3},1\right) \\ \left(\mathrm{vi}\right)x^2-2x+4y^2=0 \\ \Rightarrow \left(x^2-2x\right)+4\left(y^2\right)=0 \\ \Rightarrow \left(x^2-2x+1\right)+4\left(y^2\right)=1 \\ \Rightarrow {\left(x-1\right)}^2+4{\left(y\right)}^2=1 \\ \Rightarrow \frac{{\left(x-1\right)}^2}{1}+\frac{{\left(y\right)}^2}{{\displaystyle \frac{1}{4}}}=9 \\ \text{Centre=}\left(1,0\right) \\ \text{Major axis=2}\mathit{\text{a}} \\ =2\times 1 \\ =2 \\ \text{Minor axis=2}\mathit{\text{b}} \\ =2\times \frac{1}{2} \\ =1 \\ e=\sqrt{1-\frac{b^2}{a^2}} \\ \Rightarrow e=\sqrt{1-\frac{{\displaystyle \frac{1}{4}}}{1}} \\ \Rightarrow e=\frac{\sqrt{3}}{2} \\ \mathrm{Foci}=\left(x\pm ae,y\right) \\ =\left(1\pm \frac{3}{\sqrt{2}},0\right) \end{gathered}$$