Geo5.2.3

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Find the eccentricity,coordinates of focus,length of latus rectum and equations of directrices of the ellipse 4x^{2}+y^{2}-8x+2y+1=0\,

The given equation can be written as

4(x^{2}-2x+1)+(y^{2}+2y+1)=4\,

4(x-1)^{2}+(y+1)^{2}=4\,

Dividing the equation by 4, we have

{\frac  {(x-1)^{2}}{1}}+{\frac  {(y-(-1))^{2}}{4}}=1\,

Comparing with the form {\frac  {(x-h)^{2}}{a^{2}}}+{\frac  {(y-k)^{2}}{b^{2}}}=1\, we get

a^{2}=1,b^{2}=4,(h,k)=(1,-1),a<b\,

eccentricity is e={\sqrt  {{\frac  {4-1}{4}}}}={\frac  {{\sqrt  {3}}}{2}}\,

Focii are (h,\pm be)=(1,-1\pm {\sqrt  {3}})\,

Length of the latus rectum is {\frac  {2a^{2}}{b}}={\frac  {2\cdot 1}{2}}=1\,

Equations of directrices are y=k\pm {\frac  {b}{e}}=-1\pm {\frac  {4}{{\sqrt  {3}}}},{\sqrt  {3}}y+{\sqrt  {3}}\pm 4=0\,


Main Page:Geometry:The Ellipse