17. Length of tangent, subtangent, normal and subnormal

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 tangent = y1 cosec ψ
length of the normal = y1 sec ψ
length of the subtangent = y1 cot ψ
length of the subnormal = y1 tan ψ

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