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
$\frac{1}{\sqrt{2}}$

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

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

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

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

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Solution

The correct option is C
$3 x^{2} + 3 y^{2} + 2 x y - 2 x - 2 y + 1 = 0$

we will use the definition of conic to arrive at its equation.

$\frac{P S}{P M} = e$

P is our moving point and let its coordinate be (h,k)

PS = $\sqrt{(h - 0)^{2} + (k - 0)^{2}}$ = $\sqrt{h^{2} + k^{2}}$

PM = perpendicular distance of P from L

= $∣ ∣ \begin{matrix} \frac{h + k - 1}{\sqrt{2}} \end{matrix} ∣ ∣$

e = $\frac{1}{\sqrt{2}}$ (given)

$\frac{P S}{P M} = e \Rightarrow \sqrt{h^{2} + k^{2}} = \frac{1}{\sqrt{2}} \times ∣ ∣ \begin{matrix} \frac{h + k - 1}{\sqrt{2}} \end{matrix} ∣ ∣$

$\Rightarrow h^{2} + k^{2} = \frac{(h + k - 1)^{2}}{2^{2}}$

$\Rightarrow 4 h^{2} + 4 k^{2} = h^{2} + k^{2} - 2 h k + 1 - 2 h - 2 k$

$\Rightarrow 3 h^{2} + 3 k^{2} - 2 h k - 2 h - 2 k + 1 = 0$
we will replace (h,k) with (x,y) to get the equation of conic
$\Rightarrow 3 x^{2} + 3 y^{2} - 2 x y - 2 x - 2 y + 1 = 0$

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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 1√2

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
$\frac{1}{\sqrt{2}}$