## 1. Subnormal of ellipse

The Subnormal at a point P of the conic k is the portion z = JK of the axis cut by the projection J of P and the normal of the conic at P intersecting the axis at K.
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)2 + (y/b)2 = 1.

Points I, J are inverse with respect to the circle k of radius a, hence:
xx' = a2  ==>  x' = a2/x.
Right-angled triangles KFJ and FJI are similar hence (setting z=JK):
IJ/JF = (x'-x)/y = FJ/JK = y/JK = y/z ==>
z = y2/(x'-x) = y2/(a2/x-x) = y2x/(a2-x2) ==>
z/x = y2/(a2-x2) = ((a2b2-x2b2)/a2)/(a2-x2) = b2/a2 ==>
z = e2 x. (e: eccentricity of the ellipse).
The subnormal of the ellipse is a linear function of the ordinate x.

## 2. Subnormal of the parabola (constant)

For the parabola given in canonical form through the equation:
2py = x2.
(y = AJ, x = JP, z = JK).
The similar triangles PJK and KPI, with A (vertex of the parabola) the middle of IJ =>
x2 = (2y)z => z = x2/2y ==> z = p. (the parameter of the parabola).
The subnormal of the parabola is constant and equal to its parameter.

## 3. Subnormal of the hyperbola

Below I calculate the subnormal for the hyperbola given in the canonical form:
(x/a)2 - (y/b)2 = 1.

Points I(x'), J(x) are inverse with respect to the circle k of radius a, hence:
xx' = a2  ==>  x' = a2/x.
From the right-angled triangle KPI:
y2 = (IJ)(JK) = (x-x')z ==>
z = y2/(x-x') = y2/(x-a2/x) = y2x/(x2-a2) ==>
z/x = y2/(x2-a2) = ((x2b2-a2b2)/a2)/(x2-a2) = b2/a2 ==>
z = e2 x. (e: eccentricity of the hyperbola).
The subnormal of the hyperbola is a linear function of the ordinate x.

## 4. Remark

A unified approach can be given by taking into account that a conic is characterized by the property of its points P to have distance PF from a fixed point (focus) constant proportional to a linear function of the cartesian coordinates (x,y) [BriotBouquet, p. 184] :
PF2 = k2(ax+by+c)2.
F is the focus and ax+by+c = 0 gives the equation of the directrix with respect to F.

### Bibliography

[BriotBouquet] Briot and Bouquet Elements of Analytical Geometry Werner School book company, New York 1896