Question

Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse: 

4x² + y² − 8x + 2y + 1 = 0 

Answer 1

$$4x^2+y^2-8x+2y+1=0$$
$$\Rightarrow 4(x^2-2x)+(y^2+2y)=-1$$
$$\Rightarrow 4(x^2-2x+1)+(y^2+2y+1)=-1+4+1$$
$$\Rightarrow 4{(x-1)}^2+{(y+1)}^2=4$$
$$\Rightarrow \frac{{(x-1)}^2}{1}+\frac{{(y+1)}^2}{4}=1$$
$$\text{ Here },x_1=1a\text{ and }{y}_1=-1$$
$$\text{ Also },a=1\text{ and }b=2$$
$$\text{ Centre }=(x_1,y_1)=(1,-1)$$
$$\text{ Major axis }=2b$$
$$=2\times 2$$
$$=4$$
$$\text{ Minor axis }=2a$$
$$=2\times 1$$
$$=2$$
$$e=\sqrt{1-\frac{a^2}{b^2}}$$
$$\Rightarrow e=\sqrt{1-\frac{1}{4}}$$
$$\Rightarrow e=\frac{\sqrt{3}}{2}$$
$$\text{ Foci }=(x_1,y_1\pm be)$$
$$=(1,-1\pm \sqrt{3})$$
$$$$