# Lesson REVIEW of lessons on hyperbolas

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## REVIEW of lessons on hyperbolas

This file is your guide on my lessons on  hyperbolas  under the topic  Conic sections  in this site.  These lessons are:
- Hyperbola definition, canonical equation, characteristic points and elements;
- Hyperbola focal property;
- Tangent lines and normal vectors to a hyperbola;
- Optical property of a hyperbola.

In the first lesson  Hyperbola definition, canonical equation, characteristic points and elements  the basic notions are introduced related to hyperbolas,
such as  canonical equation of a hyperbola,  principal axes  of a hyperbola,  real semiaxis  and  imaginary semiaxis  of a hyperbola,  linear eccentricity  of a hyperbola,  foci  and  focal distance  of a hyperbola.
 In this lesson,  hyperbola is defined as a curve in a plane such that all its points  {(,)}  satisfy the equation                                                      (1) with real positive numbers   > , >  in some rectangular coordinate system  OXY  in the plane (see  Figure 1). Figure 1.  The hyperbola and its focuses

In the next lesson  Hyperbola focal property  the focal property of a hyperbola is introduced and proved.
The focal property of a hyperbola states that

A curve on a plane is an hyperbola if and only if the modulus of the difference of distances from any point of the curve
to the two fixed points (foci) on a plane is a constant
.
 The focal property of a hyperbola is the characteristic property.               Based on this property, one can define a hyperbola as a curve in a plane such that the modulus of the difference of distances from any point of the curve to the two fixed points (foci) in a plane is the constant independent from the position of the point on the curve. This definition is equivalent to the algebraic definition of hyperbolas of the lesson  Hyperbola definition, canonical equation, characteristic points and elements. So, the two definitions describe actually the same class of curves in a plane. Figure 2.  To the hyperbola focal property: the modulus of the difference of lengths   and   is the constant independent on the position of the point  M  on the hyperbola.

In the lesson  Tangent lines and normal vectors to a hyperbola  the formulas are derived for a tangent line to a hyperbola in a plane.
Tangent line to the hyperbola    at the point  (,)  has the equation  .                                   (2)
Tangent line to the hyperbola    at the point  (,)  has the equation  .        (3)
Usually, deriving this kind of formulas requires  Calculus.  In the lesson I am talking about I didn't use  Calculus.
I simply checked that the straight lines  (2)  and  (3)  pass through the given point  (,)  and have only one common point with the hyperbola.
This is enough for a straight line to be a tangent line to the smooth convex figure as a branch of a hyperbola is.

The corollaries are derived from the formulas  (2)  and  (3). They relate to the normal vector and guiding vector of a tangent line to an ellipse.
The unit "inward" normal vector to the hyperbola    at the point  (,)  is  (,),  where = .
The unit "outward" normal vector to the hyperbola    at the point  (,)  is  (,),  where = .
The guiding vector of the tangent line to the hyperbola    at the point  (,)  is  (,)  or  (,)
.

In the lesson  Optical property of a hyperbola  the optical property of a hyperbola is considered and proved.

 Optical property of a hyperbola reads as follows  (Figure 3):         If to put the source of light into one of the two hyperbola's focus points                        and if the internal surface of the hyperbola reflects the light rays as a         mirror, then all the light rays emitted by the source coincide after reflection         with the straight rays released from the second hyperbola's focus point. Figure 3  displays the hyperbola with the focus points  F1 and  F2.  A source of light is placed at the focus point  F1.  Light rays, emitted by the source, after reflecting from the hyperbola internal surface coincide with the straight rays released from the second focus points  F2. You can interchange focuses  F1 and  F2. Figure 3.  Optical property of a hyperbola
 This optical property is equivalent to any of the following geometric facts  (Figure 4): 1.  For any hyperbola's point the angles between the tangent line to the hyperbola at this    point and the straight lines drawn from the hyperbola foci to the point are congruent.         2.  For any hyperbola's point the angles between the normal to the hyperbola at this point    and the straight lines drawn from the hyperbola foci to the point are congruent. 3.  For any hyperbola's point the normal to the hyperbola at this point bisects    the angle between the straight lines drawn from the hyperbola foci to the point. 4.  For any hyperbola's point the tangent line to the hyperbola at this point bisects the Figure 4.  To the optical property of a hyperbola   (, , , )
angle between the focal vector to the point and the continuation of the other focal vector.

Figure 4  displays the hyperbola with the focus points  F1=(F,0)  and  F2=(-F,0),  where  F  is half of the focal distance. The foci are connected
with the point  M  at the hyperbola, which is chosen by an arbitrary way. The tangent line and the normal line at the point  M  are displayed too.
The optical property says that

- the angles    and   between the tangent line and the straight lines drawn from the hyperbola foci to the given point are congruent:  =;

- the angles    and   between the normal line and the straight lines drawn from the hyperbola foci to the given point are congruent:  =.

- the angle between the tangent line and the focal vector is congruent to the angle between the tangent line and the continuation of the other
focal vector:  =  and  =.  (Actually,  all four of these vectors are congruent:  ===).

The proof is fully elementary.  It is based on calculations and comparing the scalar products of the focal vectors and the normal vector.
The calculations use the explicit formulas for the focal vectors  r1  and  r2  (Figure 4),  including formulas for their lengths from the lesson
Hyperbola focal property,  as well as the explicit formula for the hyperbola's normal vector of the lesson  Tangent lines and normal vectors to a hyperbola.

Two bright spots are central in this series of lessons.  The first is the elementary proof of equivalency the algebraic and the geometric definitions of a hyperbola  (the lesson  Hyperbola focal property).  The second is the elementary proof of the optical property of a hyperbola  (the lesson  Optical property of a hyperbola).

By combining the  hyperbola focal property  and the  hyperbola optical property  we can formulate even stronger statement:

If to put two sources of light into two hyperbola's focus points, and
if the internal surface of the hyperbola reflects the light rays as a mirror, and
if the speed of light is constant for the media filling the plane, then
- all the light rays emitted by one source, after reflection at the hyperbola internal surface will coincide with the light rays
emitted by the other source, and
- the light rays emitted by the two sources at the same time moment will arrive to the hyperbola branch surface with the
time shift/delay (due to the difference of distances) constant for all the branch points
.

And,  yeah,  in order to this statement would be true,  I should probably add an assumption that the time delay produced by the reflection act is the same  (or negligeable)  for all the points at the hyperbola internal surface.

For similar lessons on ellipses see  REVIEW of lessons on ellipses  in this site.
For similar lessons on parabolas see  REVIEW of lessons on parabolas  in this site.

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