Order of differential equation

[lizzie]

Find the order of the differential equation of the following curve:
Family of parabolos having fixed directrix.

Thanks for any help.

 

[HallsofIvy]

What does the general equation of such a family look like? How many constants are there in that general formula. Generally speaking you need to differentiate once to eliminate one constant. The number of constants is the order of the equation.

 

[lizzie]

I am unable to find the general equation. Please give me some hint.

 

[tiny-tim]

I am unable to find the general equation. Please give me some hint.

Hi lizzie!

(how many dimensions is this in? and is the position of the focus constrained in any way?)

Hint: for a fixed focus and directrix, the parabola is all points whose distance from the focus equals its (perpendicular) distance from the directrix.

 

[HallsofIvy]

The vertex of a parabola is exactly half way between the focus and the directrix so a parabola having focus (0, f) and directrix y= -d has vertex at (0, (f-d)/2). The focal distance (distance from the vertex to the focus) is then f- (f-d)/2= (f+d)/2. The equation of such a parabola is y= 2(f+d)x²+ (f-d)/2. How many parameters (constants) does that involve?