Geo5.2.1

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Find the eccentricity,coordinates of focus,length of latus rectum and equations of directrices of the ellipse 9x^{2}+16y^{2}=144\,

Given ellipse 9x^{2}+16y^{2}=144\,

Rearranging the equation,we have in the form

{\frac  {9x^{2}}{144}}+{\frac  {16y^{2}}{144}}-1=0\,

{\frac  {x^{2}}{16}}+{\frac  {y^{2}}{9}}-1=0\,

From the equation

a^{2}=16,b^{2}=9,a=4,b=3\,

a>b\,

Therefore eccentricity of the ellipse is

e={\sqrt  {{\frac  {a^{2}-b^{2}}{a^{2}}}}}={\sqrt  {{\frac  {7}{16}}}}={\frac  {{\sqrt  {7}}}{4}}\,

Foci are

(\pm ae,0)=(\pm {\sqrt  {7}},0)\,

Length of the latusrectum is {\frac  {2b^{2}}{a}}={\frac  {18}{4}}={\frac  {9}{2}}\,

Equations of directrices are x=\pm {\frac  {a}{e}}=\pm {\frac  {16}{{\sqrt  {7}}}}\,

{\sqrt  {7}}x=\pm 16\,

Main Page:Geometry:The Ellipse