The parabola with directrix $x+2 y-1=0$ and focus $(1,0)$ is
The parabola with directrix $x+2 y-1=0$ and focus $(1,0)$ is
$4 x^2-4 x y+y^2-8 x+4 y+4=0$
$4 x^2+4 x y+y^2-8 x+4 y+4=0$
$4 x^2+5 x y+y^2+8 x-4 y+4=0$
$4 x^2-4 x y+y^2-8 x-4 y+4=0$
Solution
Let $P(x, y)$ be any point on the parabola
By definition of Parabola $P M=P S$
$\Rightarrow \quad \frac{x+2 y-1}{\sqrt{1+4}}=\sqrt{(x-1)^2+y^2}$
On squaring both sides, we get
$\begin{gathered}
x^2+4 y^2+1+4 x y-4 y-2 x \\
=5\left(x^2+1-2 x+y^2\right) \\
\Rightarrow \quad 4 x^2+y^2-8 x+4 y-4 x y+4=0
\end{gathered}$