Editing Parabola (section)

== Conic section and quadratic form ==
=== Diagram, description, and definitions ===
[[image:Parabolic conic section.svg|thumb|Cone with cross-sections]]

The diagram represents a [[cone]] with its axis {{overline|AV}}. The point A is its [[apex (geometry)|apex]]. An inclined [[Cross section (geometry)|cross-section]] of the cone, shown in pink, is inclined from the axis by the same angle {{mvar|θ}}, as the side of the cone. According to the definition of a parabola as a conic section, the boundary of this pink cross-section EPD is a parabola.

A cross-section perpendicular to the axis of the cone passes through the vertex P of the parabola. This cross-section is circular, but appears [[Ellipse|elliptical]] when viewed obliquely, as is shown in the diagram. Its centre is V, and {{overline|PK}} is a diameter. We will call its radius&nbsp;{{mvar|r}}.

Another perpendicular to the axis, circular cross-section of the cone is farther from the apex A than the one just described. It has a [[Chord (geometry)|chord]] {{overline|DE}}, which joins the points where the parabola [[Intersection (Euclidean geometry)|intersects]] the circle. Another chord {{overline|BC}} is the [[perpendicular bisector]] of {{overline|DE}} and is consequently a diameter of the circle. These two chords and the parabola's axis of symmetry {{overline|PM}} all intersect at the point&nbsp;M.

All the labelled points, except D and E, are [[coplanar]]. They are in the plane of symmetry of the whole figure. This includes the point F, which is not mentioned above. It is defined and discussed below, in {{slink||Position of the focus}}.

Let us call the length of {{overline|DM}} and of {{overline|EM}} {{mvar|x}}, and the length of {{overline|PM}}&nbsp;{{mvar|y}}.

=== Derivation of quadratic equation ===
The lengths of {{overline|BM}} and {{overline|CM}} are:
{{unbulleted list | style = padding-left:1.6em;
| <math>\overline\mathrm{BM} = 2y\cos\theta</math>{{pad|1em}}(triangle BPM is [[isosceles]], because <math>\overline{PM} \parallel \overline{AC} \implies \angle PMB = \angle ACB = \angle ABC</math>
| <math>\overline\mathrm{CM} = 2r</math>{{pad|1em}}(PMCK is a [[parallelogram]]).
}}
Using the [[Chord theorem|intersecting chords theorem]] on the chords {{overline|BC}} and {{overline|DE}}, we get
<math display="block">\overline\mathrm{BM} \cdot \overline\mathrm{CM} = \overline\mathrm{DM} \cdot \overline\mathrm{EM}.</math>

Substituting:
<math display="block">4ry\cos\theta = x^2.</math>

Rearranging:
<math display="block">y = \frac{x^2}{4r\cos\theta}.</math>

For any given cone and parabola, {{mvar|r}} and {{mvar|θ}} are constants, but {{mvar|x}} and {{mvar|y}} are variables that depend on the arbitrary height at which the horizontal cross-section BECD is made. This last equation shows the relationship between these variables. They can be interpreted as [[Cartesian coordinate system|Cartesian coordinates]] of the points D and E, in a system in the pink plane with P as its origin. Since {{mvar|x}} is squared in the equation, the fact that D and E are on opposite sides of the {{mvar|y}} axis is unimportant. If the horizontal cross-section moves up or down, toward or away from the apex of the cone, D and E move along the parabola, always maintaining the relationship between {{mvar|x}} and {{mvar|y}} shown in the equation. The parabolic curve is therefore the [[Locus (mathematics)|locus]] of points where the equation is satisfied, which makes it a [[Graph of a function|Cartesian graph]] of the quadratic function in the equation.

=== Focal length ===
It is proved in a [[#In a cartesian coordinate system|preceding section]] that if a parabola has its vertex at the origin, and if it opens in the positive {{mvar|y}} direction, then its equation is {{math|1=''y'' = {{sfrac|''x''<sup>2</sup>|4''f''}}}}, where {{mvar|f}} is its focal length.{{efn|As stated above in the lead, the focal length of a parabola is the distance between its vertex and focus.}} Comparing this with the last equation above shows that the focal length of the parabola in the cone is {{math|''r'' sin ''θ''}}.

=== Position of the focus ===
In the diagram above, the point&nbsp;V is the [[Perpendicular#Foot of a perpendicular|foot of the perpendicular]] from the vertex of the parabola to the axis of the cone. ''The point&nbsp;F is the foot of the perpendicular from the point&nbsp;V to the plane of the parabola.''{{efn|The point&nbsp;V is the centre of the smaller circular cross-section of the cone. The point&nbsp;F is in the (pink) plane of the parabola, and the line {{overline|VF}} is perpendicular to the plane of the parabola.}} By symmetry, F is on the axis of symmetry of the parabola. Angle VPF is [[Complementary angles|complementary]] to {{mvar|θ}}, and angle PVF is complementary to angle VPF, therefore angle PVF is {{mvar|θ}}. Since the length of {{overline|PV}} is {{mvar|r}}, the distance of F from the vertex of the parabola is {{math|''r'' sin ''θ''}}. It is shown above that this distance equals the focal length of the parabola, which is the distance from the vertex to the focus. The focus and the point&nbsp;F are therefore equally distant from the vertex, along the same line, which implies that they are the same point. Therefore, ''the point&nbsp;F, defined above, is the focus of the parabola''.

This discussion started from the definition of a parabola as a conic section, but it has now led to a description as a graph of a quadratic function. This shows that these two descriptions are equivalent. They both define curves of exactly the same shape.

=== Alternative proof with Dandelin spheres ===
[[image:Dandelin-parabel.svg|thumb|Parabola (red): side projection view and top projection view of a cone with a Dandelin sphere]]
An alternative proof can be done using [[Dandelin spheres]]. It works without calculation and uses elementary geometric considerations only (see the derivation below).

The intersection of an upright cone by a plane <math>\pi</math>, whose inclination from vertical is the same as a [[generatrix]] (a.k.a. generator line, a line containing the apex and a point on the cone surface) <math>m_0</math> of the cone, is a parabola (red curve in the diagram).

This generatrix <math>m_0</math> is the only generatrix of the cone that is parallel to plane <math>\pi</math>. Otherwise, if there are two generatrices parallel to the intersecting plane, the intersection curve will be a [[hyperbola]] (or [[Degenerate conic|degenerate hyperbola]], if the two generatrices are in the intersecting plane). If there is no generatrix parallel to the intersecting plane, the intersection curve will be an [[ellipse]] or a [[circle]] (or [[Degenerate conic|a point]]).

Let plane <math>\sigma</math> be the plane that contains the vertical axis of the cone and line <math>m_0</math>. The inclination of plane <math>\pi</math> from vertical is the same as line <math>m_0</math> means that, viewing from the side (that is, the plane <math>\pi</math> is perpendicular to plane <math>\sigma</math>), <math>m_0 \parallel \pi</math>.

In order to prove the directrix property of a parabola (see {{slink||Definition as a locus of points}} above), one uses a [[Dandelin spheres|Dandelin sphere]] <math>d</math>, which is a sphere that touches the cone along a circle <math>c</math> and plane <math>\pi</math> at point <math>F</math>. The plane containing the circle <math>c</math> intersects with plane <math>\pi</math> at line <math>l</math>. There is a [[Reflection symmetry|mirror symmetry]] in the system consisting of plane <math>\pi</math>, Dandelin sphere <math>d</math> and the cone (the [[plane of symmetry]] is <math>\sigma</math>).

Since the plane containing the circle <math>c</math> is perpendicular to plane <math>\sigma</math>, and <math>\pi \perp \sigma</math>, their intersection line <math>l</math> must also be perpendicular to plane <math>\sigma</math>. Since line <math>m_0</math> is in plane <math>\sigma</math>, <math>l \perp m_0</math>.

It turns out that <math>F</math> is the ''focus'' of the parabola, and <math>l</math> is the ''directrix'' of the parabola.

# Let <math>P</math> be an arbitrary point of the intersection curve.
# The [[generatrix]] of the cone containing <math>P</math> intersects circle <math>c</math> at point <math>A</math>.
# The line segments <math>\overline{PF}</math> and <math>\overline{PA}</math> are tangential to the sphere <math>d</math>, and hence are of equal length.
# Generatrix <math>m_0</math> intersects the circle <math>c</math> at point <math>D</math>. The line segments <math>\overline{ZD}</math> and <math>\overline{ZA}</math> are tangential to the sphere <math>d</math>, and hence are of equal length.
# Let line <math>q</math> be the line parallel to <math>m_0</math> and passing through point <math>P</math>. Since <math>m_0 \parallel \pi</math>, and point <math>P</math> is in plane <math>\pi</math>, line <math>q</math> must be in plane <math>\pi</math>. Since <math>m_0 \perp l</math>, we know that <math>q \perp l</math> as well.
# Let point <math>B</math> be ''the foot of the perpendicular'' from point <math>P</math> to line <math>l</math>, that is, <math>\overline{PB}</math> is a segment of line <math>q</math>, and hence <math>\overline{PB} \parallel \overline{ZD}</math>.
# From [[intercept theorem]] and <math>\overline{ZD} = \overline {ZA}</math> we know that <math>\overline{PA} = \overline {PB}</math>. Since <math>\overline{PA} = \overline {PF}</math>, we know that <math>\overline{PF} = \overline {PB}</math>, which means that the distance from <math>P</math> to the focus <math>F</math> is equal to the distance from <math>P</math> to the directrix <math>l</math>.