what is the lenght of subtangent in the term of differential quotients of any general curve...

Let tangent and normal to a parabola at a point P(x1,y1) be extended to meet the axis of the parabola at N and T respectively.

PT is termed the length of the tangent.
PN is termed the length of the normal.

Drop a perpendicular to the axis from the point P and call it PP'.

P' T = subtangent
P’ N = subnormal

If the tangent makes an angle of ψ with the axis


length of the subtangent = y1 cot ψ = y1/(dy/dx)

dy/dx =tan ψ = 2a/y1 = m (slope of the tangent)