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The equation of an ellipse whose focus is \left ( 1,1 \right ), whose directrix is\mathrm{x-y+3=0} and whose eccentricity is \frac{1}{2}, is given by

Option: 1

\mathrm{7 x^2+2 x y+7 y^2+10 x-10 y+7=0}


Option: 2

\mathrm{7 x^2-2 x y+7 y^2-10 x+10 y+7=0}


Option: 3

\mathrm{7 x^2-2 x y+7 y^2-10 x-10 y-7=0}


Option: 4

\mathrm{7 x^2-2 x y+7 y^2+10 x+10 y-7=0}


Answers (1)

best_answer

Let any point on it be \mathrm{\left ( x,y \right )}

then by definition,


\mathrm{\sqrt{(x+1)^2+(y-1)^2}=\frac{1}{2}\left|\frac{x-y+3}{\sqrt{1^2+1^2}}\right|}
Squaring and simplifying, we get
\mathrm{7 x^2+2 x y+7 y^2+10 x-10 y+7=0} 

which is the required ellipse.

 

 

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