Eccentricity of a Conic

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Q.

The equation of directrix of a conic is
$L : x + y - 1 = 0$ and the focus is the point $(0, 0)$ . Find the equation of the conic if its eccentricity is
$\frac{1}{\sqrt{2}}$

$3 x^{2} + 3 y^{2} + 2 x y + 2 x + 2 y - 1 = 0$
$3 x^{2} + 3 y^{2} + 2 x y + 2 x + 2 y + 1 = 0$
$3 x^{2} + 3 y^{2} + 2 x y - 2 x - 2 y + 1 = 0$
$3 x^{2} - 3 y^{2} + 2 x y + 2 x - 2 y + 1 = 0$

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Q.

P is a moving point, S is the focus and L is the directrix as shown in figure.

Which of the following represents the equation of a conic?

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