Geo5.2.4

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Find the eccentricity,coordinates of focus,length of latus rectum and equations of directrices of the ellipse x^{2}+2y^{2}-4x+12y+14=0\,

The given equation can be written as

(x^{2}-4x+4)+2(y^{2}+6y+9)=8\,

(x-2)^{2}+2(y+3)^{2}=8\,

Dividing the equation by 8, we have

{\frac  {(x-2)^{2}}{8}}+{\frac  {(y-(-3))^{2}}{4}}=1\,

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

a^{2}=8,b^{2}=4,a={\sqrt  {8}}=2{\sqrt  {2}},b=2,(h,k)=(2,-3),a>b\,

Length of major axis and minor axis are 4{\sqrt  {2}},4\,

eccentricity is e={\sqrt  {{\frac  {8-4}{8}}}}={\frac  {1}{{\sqrt  {2}}}}\,

Focii are (h\pm ae,k)=(2\pm 2,-3)=(0,-3),(4,-3)\,

Length of the latus rectum is {\frac  {2b^{2}}{a}}={\frac  {2\cdot 4}{2{\sqrt  {2}}}}=2{\sqrt  {2}}\,

Equations of directrices are x=h\pm {\frac  {a}{e}}=2\pm 2\cdot 2

x=6,x=-2\,


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