Question

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
$\text{eccentricity is}\frac{1}{\sqrt{2}}$

Answer 1

Given,
Directrx: $L:x+y-1=0$
Focus $\equiv (0,0)$
$e=\frac{1}{\sqrt{2}}$

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

$\frac{PS}{PM}=e$

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

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

$PM=$ perpendicular distance of P from Line L

$\Rightarrow PM=\left|\begin{array}{c}\frac{h+k-1}{\sqrt{2}}\end{array}\right|$

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

$\Rightarrow \frac{PS}{PM}=e\Rightarrow PS=e\times PM\Rightarrow \sqrt{h^2+k^2}=\frac{1}{\sqrt{2}}\times \left|\begin{array}{c}\frac{h+k-1}{\sqrt{2}}\end{array}\right|$

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

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

$\Rightarrow 3h^2+3k^2-2hk-2h-2k+1=0$
we will replace $(h,k)\text{with}(x,y)$
to get the equation of conic
$\Rightarrow 3x^2+3y^2+2xy-2x-2y+1=0$
Hence, the correct answer is Option c.