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The equation of an ellipse whose eccentricity $\mathrm{e=\frac{1}{2}}$ , focus is $(-1,1),$ and directrix is $\mathrm{x+y+3=0}$ is $\mathrm{7 x^2-2 x y+7 y^2+10 x+k y+7=0}$ , where $\mathrm{k=}$

Option: 1
$22$

Option: 2
$-22$

Option: 3
$10$

Option: 4
$-10$

Answers (1)

Let $\mathrm{P\left(x_1, y_1\right)}$ be a point on the ellipse.
The focus is $\mathrm{S(-1,1)}$ and the directrix is $\mathrm{x+y+3=0}$ Then, by definition of ellipse
$\mathrm{ P S=e P M }$

$\mathrm{ Or \left(x_1+1\right)^2+\left(y_1-1\right)^2=\frac{1}{4}\left(\frac{x_1+y_1+3}{\sqrt{2}}\right)^2 }$

$\mathrm{Or \: 8\left(x_1^2+y_1^2+2 x_1-2 y_1+1+1\right)=x_1^2+y_1^2+2 x_1 y_1+6 x_1+6 y_1+9}$

The locus of $\mathrm{P\left(x_1, y_1\right)}$ is

$\mathrm{ 7 x^2-2 x y+7 y^2+10 x-22 y+7=0 }$

$\mathrm{ k=-22 }$

Hence option 2 is correct.

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Kshitij

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