Geo5.2.8

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Find the equation of the ellipse with a focus at(1,-1),e=2/3 and directrix is x+y+2=0\,

Let P(x,y) be a point on the ellipse.

Focus is S(1,-1).

Then by definition, SP=ePM\, (PM is the perpendicular distance from the point to

directrix)

The equation of the ellipse is

SP^{{2}}=e^{2}PM^{{2}}\,

(x-1)^{2}+(y+1)^{2}={\frac  {4}{9}}[{\frac  {(x+y+2)^{2}}{2}}]\,

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

7x^{2}+7y^{2}-26x+10y-4xy+10=0\,


Main Page:Geometry:The Ellipse