y=4x^2+7x-1

let coordinates of A be (x1,y1)

coordinates of B be (x2,y2)

since (0,0) is the midpoint of line segment AB therfor

(x1+x2)/2=0

x1=-x2

similarly y1=-y2

now according to the equation

y1^2=4(x1)^2+7x1-1 and

y2^2=4(x2)^2+7x2-1

since y1=-y2 therefor

4(x1)^2+7x1-1 +4(x2)^2+7x2-1 =0

now after putting x1=-x2 we get

8(x2)^2=2 therefor

x2=+-(1/2)

let x2=(1/2) so x1=(-1/2)

put these value in equation and obtain value of y1 and y2

so A=(-1/2,-7/2) and B=(1/2,7/2)

using distance formula we get distance between them=sqrt[50]