Find the equation of the parabola whose focus is the origin and whose directrix is the line 2x + y – 1 = 0 

Focus S(0,0) and directrix 2x + y – 1 = 0

Let P(x, y) be any point on the parabola.

By definition: SP = PM

⇒ SP² = PM²

SP² = x² + y²

PM² = (2x + y – 1)² / (√(2² + 1))² = (2x + y – 1)² / 5

Therefore:

x² + y² = (2x + y – 1)² / 5

⇒ 5x² + 5y² = 4x² + y² + 1 + 4xy – 2y – 4x

⇒ x² + 4y² – 4xy + 2y + 4x – 1 = 0