Below I calculate the subnormal for the ellipse, the hyperbola and the parabola,

given in the canonical form.

First the case of the ellipse (x/a)

Points I, J are inverse with respect to the circle k of radius a, hence:

xx' = a

Right-angled triangles KFJ and FJI are similar hence (setting z=JK):

IJ/JF = (x'-x)/y = FJ/JK = y/JK = y/z ==>

z = y

z/x = y

z = e

The subnormal of the ellipse is a linear function of the ordinate x.

2py = x

(y = AJ, x = JP, z = JK).

The similar triangles PJK and KPI, with A (vertex of the parabola) the middle of IJ =>

x

The subnormal of the parabola is constant and equal to its parameter.

(x/a)

Points I(x'), J(x) are inverse with respect to the circle k of radius a, hence:

xx' = a

From the right-angled triangle KPI:

y

z = y

z/x = y

z = e

The subnormal of the hyperbola is a linear function of the ordinate x.

PF

F is the focus and ax+by+c = 0 gives the equation of the directrix with respect to F.

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