ConicSubnormal

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

http://users.math.uoc.gr/~pamfilos/