Question

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

Answer 1

Focus(0,0)

$e{q}^n$ of directrix = $2x-y-1=0$

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

$\Rightarrow (SA{)}^2=(AM{)}^2$

$\Rightarrow (x-0{)}^2+(y-0{)}^2=(\frac{2x-y-1}{\sqrt{2^2+(-1{)}^2}}{)}^2$

$\Rightarrow x^2+y^2=\frac{(2x-y-1{)}^2}{5}$

$\Rightarrow 5x^2+5y^2=4x^2+y^2-4xy+1+4x-2y$

$\Rightarrow x^2+4y^2+4xy-4x+2y-1=0$

Which is the required $e{q}^n$ of parabola.