Find the equation of parabola whose
Focus is $(0, 0)$ and the directrix is $2 x - y - 1 = 0$

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Solution

Focus(0,0)
$e q^{n}$ of directrix = $2 x - y - 1 = 0$
Let A(x,y) be the pt. on the parabola
and AM be the length of the perpendicular
from A to the directrix.
Then,
SA = AM
$\Rightarrow (S A)^{2} = (A M)^{2}$
$\Rightarrow (x - 0)^{2} + (y - 0)^{2} = (\frac{2 x - y - 1}{\sqrt{2^{2} + (- 1)^{2}}})^{2}$
$\Rightarrow x^{2} + y^{2} = \frac{(2 x - y - 1)^{2}}{5}$
$\Rightarrow 5 x^{2} + 5 y^{2} = 4 x^{2} + y^{2} - 4 x y + 1 + 4 x - 2 y$
$\Rightarrow x^{2} + 4 y^{2} + 4 x y - 4 x + 2 y - 1 = 0$
Which is the required $e q^{n}$ of parabola.

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