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# The equation of the ellipse whose focus is $(1,-1)$,the directrix the line $x-y-3=0$ and eccentricity $\large\frac{1}{2}$ is

$\begin{array}{1 1}7x^2+2xy+7y^2+10x-10y+7=0\\7x^2+2xy+7y^2+7=0\\7x^2+2xy+7y^2+10x-10y-7=0\\7x^2-2xy-7y^2-7=0\end{array}$

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• For an ellipse the ratio $\large\frac{SP}{PM}$$=e where e < 1 and S(ae,0) is the focus and P(x,y) is any point on the ellipse and PM is the perpendicular to the directrix. Answer : 7x^2+2xy+7y^2+10x-10y+7=0 Given focus is (1,-1) and equation of the directrix is x-y=3 Eccentricity e=\large\frac{1}{2} Hence SP=ePM SP^2=e^2PM^2 (ie) SP^2=\large\frac{1}{4}$$(PM)^2$
$4SP^2=PM^2$
(ie) $4[(x+1)^2+(y-1)^2]=\big(\large\frac{x-y-3}{\sqrt{1^2+(-1)^2}}\big)^2$
$\Rightarrow 8(x^2+y^2+2x-2y+2)=(x-y+3)^2$
On simplifying we get,
$7x^2+2xy+7y^2+10x-10y+7=0$

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The equation of ellipse whose focus is (-1,1) the directrix the line x-y+3=0 and eccentricity 1 by 2