prove the parablola

Question

prove the parablola with foucus (c,0) and directricx x =-c has equation y^2 = 4cx

Answer 1

The distance from the focus to a point on the parabola is equal to the distance from the directrix to the same point on the parabola.

 

Let (x,y) be a point on the parabola.  A horizontal line will connect the (x,y) to the closest point on the directrix, (-c,y).

 

√[(x - c)² + (y - 0)²] = √[(x - (-c))² + (y - y)²]

 

√[(x - c)² + y²] = √[(x + c)²]

 

x² - 2cx + c² + y² = x² + 2cx + c²

 

y² = 4cx