Geo5.2.5

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Find the lengths of major axis,minor axis,eccentricity,coordinates of focus,length of latus rectum and equations of directrices of the ellipse 3x^{2}+y^{2}-6x-2y-5=0\,

The given equation can be written as

3(x^{2}-2x+1)+(y^{2}-2y+1)=9\,

3(x-1)^{2}+(y-1)^{2}=9\,

Dividing the equation by 9, we have

{\frac  {(x-1)^{2}}{3}}+{\frac  {(y-1))^{2}}{9}}=1\,

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

a^{2}=3,b^{2}=9,a={\sqrt  {3}},b=3,(h,k)=(1,1),b>a\,

Length of major axis and minor axis are 2{\sqrt  {3}},6\,

eccentricity is e={\sqrt  {{\frac  {9-3}{9}}}}={\sqrt  {{\frac  {2}{3}}}}\,

Focii are (h,k\pm be)=(1,1\pm {\sqrt  {6}})\,

Length of the latus rectum is {\frac  {2a^{2}}{b}}={\frac  {2}{3}}\,

Equations of directrices are y=k\pm {\frac  {b}{e}}=1\pm {\frac  {3{\sqrt  {3}}}{{\sqrt  {2}}}}

{\sqrt  {2}}y={\sqrt  {2}}\pm 3{\sqrt  {3}}\,


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