Minimum distance between two parabolas

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Charlesss

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Question) find the minimum distance between g(x)=Ax^2+BX+C and f(x)= ax^2+bx+c (the 2 parabolas are not intersecting)

This is as far as i got ........

d= sqrt (x_1 - x_2)^2 + [(Ax^2 + Bx + C) - ( ax^2 + bx + c)]

d^2 = h(x) = (x_1 - x_2)^2 + x^2(A-a) + x(B-b) + (C-c)

h'(x) = 2x(A-a) + (B-b) = 0

x= (B-b)/2(A-a)

(x,g(x)) and (x,f(x)) are the 2 closest points on the 2 parabolas.

I showed this to my teacher and he told me it was wrong but im not sure where..
 
Charlesss said:
Question) find the minimum distance between g(x)=Ax^2+BX+C and f(x)= ax^2+bx+c

This is as far as i got ........

d= sqrt (x_1 - x_2)^2 + [(Ax^2 + Bx + C) - ( ax^2 + bx + c)]<<< What is x_1, x_2 and x?

Clearly write out your assumptions and do not skip steps. For example, I think, you started the following way.

Let the points under consideration are

(x_1, g(x_1) and (x_2, f(x_2))

d^2 = h(x_1, x_2) = (x_1 - x_2)^2 + (Ax_1^2 +Bx_1 + C - ax_2^2 - bx_2 -c)^2

So h is a function of two variables - x_1 and x_2.



d^2 = h(x) = (x_1 - x_2)^2 + x^2(A-a) + x(B-b) + (C-c)

h'(x) = 2x(A-a) + (B-b) = 0

x= (B-b)/2(A-a)

(x,g(x)) and (x,f(x)) are the 2 closest points on the 2 parabolas.

I showed this to my teacher and he told me it was wrong but im not sure where..
Click to expand...
 
x_1 and x_2 is x subscript 1 and x subscript 2 respectively. i wasnt sure were to begin so i just put in as much information into the distance formula as i could.
 
First, if the two parabola intersect the distance between is zero.
Then if they do not intersect, then the minimum distance occurs at points where they have parallel tangents.
Thus at those points the have the same slope: \(\displaystyle 2Ax_1 + B = 2ax_2 + b\).
You need to find: \(\displaystyle \left( {x_1 ,f(x_1 )} \right)\,\& \,\left( {x_2 ,g(x_2 )} \right)\).
You also need to find conditions under which the two intersect.
 
i was told the parabolas dont intersect and arent there a million tangent points that are parallel between the 2 parabolas im not sure where to begin with the pointers u gave me pka
 
so by simply solving for x from the tangent line equation i should be ale to get the x part of the distance formula am i right in this assumption?
 
The point is: minimize the distance between those points.
 
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