Geo5.2.10

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Find the equation of the ellipse referred to its major axis and minor axis as the axes of coordinates a and y axes respectively with latus rectum of length 4 and distance between foci 4{\sqrt  {2}}\,

Let the equation be {\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}=1\, (a > b)

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

Distance between focii is 2ae=4{\sqrt  {2}},ae=2{\sqrt  {2}}\,

But b^{2}=a^{2}(1-e^{2})\,

a^{2}-2a-8=0,(a-4)(a+2)=0\,

a=4,b^{2}=2a=8\, (Since a can't be negative)

{\frac  {x^{2}}{16}}+{\frac  {y^{2}}{8}}=1\,

x^{2}+2y^{2}=16\,


Main Page:Geometry:The Ellipse