Download Geometry classwork1 September 16

September 16, 2012
Math 10
Geometry.
Geometry 9 recap: curves of the second degree.
Definition of ellipse, hyperbola and parabola: Directrix and focus
Parabola is the locus of points such that the ratio of the distance to a
given point (focus) and a given line (directrix) equals 1.
Ellipse can be defined as the
locus of points P for which the
distance to a given point (focus
F2) is a constant fraction of the
perpendicular distance to a given
line, called the directrix,
|PF2|/|PD| = e < 1.
Hyperbola can also be defined as
the locus of points for which the
ratio of the distances to one
focus and to a line (called the directrix) is a constant e. However, for a
hyperbola it is larger than 1, |PF2|/|PD| = e > 1. This constant is the
eccentricity of the hyperbola. By symmetry a hyperbola has two
directrices, which are parallel to the conjugate axis and are between it
and the tangent to the hyperbola at a vertex.
In order to show that the above definitions indeed those of an ellipse
and a hyperbola, let us obtain relation between the x and y coordinates
of a point P (x,y) satisfying the definition. Using axes shown in the
Figure, with focus F2 on the X axis at a distance l from the origin and
choosing the Y-axis for the directrix, we have
Finally, we thus obtain,
Which is the equation of an ellipse for
for
. In each case the center is at
, and the semi-axes are
and
and of a hyperbola
and
, which brings the
equation to a canonical form,
We also obtain the following relations between the eccentricity e and
the ratio of the semi-axes, a/b:
, or,
, where
plus and minus sign correspond to the case of a hyperbola and an
ellipse, respectively.
Optical property of ellipse, parabola and hyperbola
If a ray of light is reflected in a mirror, then the
reflection angle equals the incidence angle. This is
related to the Fermat principle, which states that the
light always travels along the shortest path. It is
clear from the Figure that of all reflection points P on
the line l (mirror) the shortest path between points F1
and F2 on the same side of it is such that points F1, P,
and the reflection of F2 in l, F’2, lie on a straight line.
The optical property of the ellipse. Suppose a
line l is tangent to an ellipse at a point P. Then l is
the bisector of the exterior angle F1PF2 (and its
perpendicular at point P is the bisector of F1PF2).
A light ray passing through one focus of an
elliptical mirror will pass through another focus
upon reflection.
Proof. Let X be an arbitrary point of l different from P. Since X is
outside the ellipse, we have XF1 +XF2 > PF1 +PF2, i.e., of all the points of
l the point P has the smallest sum of the distances to F1 and F2. This
means that the angles formed by the lines PF1 and PF2 with l are equal.
The optical property of the parabola. Suppose
a line l is tangent to a parabola at a point P. Let P’
be the projection of P to the directrix. Then l is
the bisector of the angle FPP’ (see figure).
If a point light source, such as a small light bulb, is
placed in the focus of a parabolic mirror, the
reflected light forms plane-parallel beam
(spotlights).
Proof. Let point P belong to a parabola and l’ be a bisector of the angle
FPP’, where |PP’| is the distance to the directrix l. Then, for any point
Q on l’, |FQ| = |QP’| ≥ |QQ’|. Hence, all points Q on l, except for Q = P,
are outside the parabola, so l’ is tangent to the parabola at point P.
Consider the following problem. Given two lines, l and l’,
P’
and a point F not on any of those lines, find a point P on
l
l such that the (signed) difference of distances from
it to l’ and F is maximal (#2 in the homework). As seen
L’
l’
in the figure, for any P’ on l the distance to l’, |P’L’| ≤
|P’L| ≤ |P’F| + |FL|, where |FL| is the distance from F to l’. Hence, |P’L’|
- |P’F| ≤ |FL| and the difference is largest (=|FL|) when point P belongs
to the perpendicular FL from point F to l’.
The optical property of the hyperbola. Suppose a
line l is tangent to a hyperbola at a point P; then l is
the bisector of the angle F1PF2, where F1 and F2 are
the foci of the hyperbola (see figure).
Proof. Let point P belong to a hyperbola and l’ be a
bisector of the angle F1PF2. Let F1’ be the reflection of
F1 in l’. Then, for any point Q on l’, |QF1| = |QF1’|, and
|QF2| - |QF1| = |QF2| - |QF1’| ≤ |F2F1’|, again by the triangle inequality.
Hence, all points Q on l’, except for Q = P, are in-between the branches
of the hyperbola, so l’ is tangent to the hyperbola at point P.
Consider the following problem. Given line l and points F1
and F2 lying on different sides of it, find point P on the
line l such that the absolute value of the difference in
distances from P to points F1 and F2 is maximal (#1 in
the homework). As above, let F2’ be the reflection of F2
in l. Then for any point X on l, |XF1| - |XF2| ≤ |F1F2’|.
P
F
L
Additional properties of ellipse, parabola and hyperbola
The optical property yields elementary proofs of some amazing results.
The isogonal property of conics. From any
point P outside an ellipse draw two tangents
to the ellipse, with tangency points X and Y.
Then the angles F1PX and F2PY are equal (F1
and F2 are the foci of the ellipse).
Proof. Let F1’, F2’ be the reflections of F1
and F2 in PX and PY, respectively (see
Figure).
Then PF1’ = PF1 and PF2’ = PF2. Moreover, the points F1, Y and F2 lie on a
line (because of the optical property). The same is true for the points
F2, X and F1’. Thus F2F1’ = F2X + XF1 = F2Y + YF1 = F2’F1. Thus, the
triangles PF2F1’ and PF1F2’ are equal (having three equal sides).
Therefore, ∠F2PF1 + 2∠F1PX =∠F2PF1’ = ∠F1P F2’= ∠F1PF2 + 2∠F2PY.
Hence ∠ F1PX =∠ F2PY, which is the desired result.
Consequence 1. The line F1P is the bisector
of the angle XF1Y.
A similar property holds for the hyperbola
Proof follows the same reasoning as for the
ellipse. Two cases should be considered
separately: when the tangency points X and
Y are either on different branches of the
hyperbola, or on the same branch, as in the
Figure.
Consequence 2. The locus of points from which a given ellipse is seen
at a right angle (i.e., the tangents to the ellipse drawn from such a
point are perpendicular) is a circle centered at the center of the
ellipse (see Figure).
Proof. Let F1, F2 be the foci of the ellipse
and suppose that the tangents to the
ellipse at X and Y intersect in P. Let F1’ be
the reflection of F1 in PX (see Figure). Then
∠XPY =∠F1’PF2 and F1’F2 = F1X+F2X, i.e., the
length of the segment F1’F2 equals the
major axis of the ellipse (the length of the
rope tying the goat). The angle F1’PF2 is
right if and only if |F1’P|2 +|F2P|2 = | F1’F2|2
(by the Pythagorean theorem). Therefore
XPY is a right angle if and only if |F1P|2 + |F2P|2 equals the square of
the major axis of the ellipse, (2a)2. But it is not difficult to see that
this condition defines a circle. Indeed, suppose F1 has Cartesian
coordinates (x1, y1), and F2 has coordinates (x2, y2). Then the
coordinates of the desired points P (x, y) satisfy the condition
But since the coefficients of x2 and y2 are equal (to 2) and the
coefficient of xy is zero, the set of points satisfying this condition is a
circle. By virtue of symmetry, its center is the midpoint of the
segment F1F2.
For the hyperbola such a circle does not always exist. When the angle
between the asymptotes of the hyperbola is acute, the radius of the
circle is imaginary. If the asymptotes are perpendicular, then the
circle degenerates into the point which is the center of the hyperbola.
Method of coordinates recap: vectors, translations.
A vector is a quantity that has magnitude (length) and
direction. This represents a translation. A vector is
represented by a directed line segment, which is a
segment with an arrow at one end indicating the
direction of movement. Unlike a ray, a directed line
segment has a specific length.
The direction is indicated by an arrow pointing from
the tail (the initial point) to the head (the terminal
point). If the tail is at point A and the head is at point
B, the vector from A to B is written as:
notation:
. Vectors may also be labeled as a single
letter with an arrow,
, or a single bold face
letter, such as vector v.
The length (magnitude) of a vector v is written |v|. Length is always a
non-negative real number.
C
Addition (subtraction?) of vectors.
One can formally define an operation of
addition on the set of all vectors (in the
space of vectors). For any two vectors,
and
, such an operation
results in a third vector,
, such that
three following rules hold,
A
B
C
A
B
(1)
(2)
(3)
D
Usually (but not necessarily always – not true for the so-called
“pseudo-vectors”), vector addition also satisfies the fourth property,
,
(4)
which allows to define the subtraction of vectors. Addition and
subtraction of vectors can be simply illustrated by considering the
consecutive translations. They allow decomposing any vector into a
sum, or a difference, of a number (two, or more) of other vectors.
The coordinate representation of vectors.
Let vector a define translation A->B under which an arbitrary point A
with coordinates (xA,yA) on the (X,Y) coordinate plane is displaced to a
point B with coordinates (xB,yB) = (xA+ax,yA+ay) on this plane. Then,
vector a is fully determined by the pair of numbers, (ax, ay), which
specify displacements along X and Y axis, respectively. If O is the
point of origin in the coordinate plane, (xO,yO) = (0, 0), it is clear, that
,
(5)
from where follows the coordinate notation of vectors,
= (xA,yA),
= (xB,yB), = (ax, ay) = a. Since numbers ax and ay denote magnitudes
of translation along the X and Y axes, respectively, corresponding to
the displacement by a vector a, it can be represented as a sum of
these two translations, a = ax ex + ay ey , or, (ax, ay) =(ax, 0) + (0, ay), or,
,
where ex =
and ey =
(6)
are vectors of length 1, called unit vectors.
The length (magnitude) of a vector, in coordinate representation, is
, a = |a| =
.
(7)
As you can see in the diagram at the right, the
length of a vector can be found by forming a
right triangle and utilizing the Pythagorean
Theorem or by using the Distance Formula.
The vector at the right translates 6 units to the
right and 4 units upward. The magnitude of the
vector is
from the Pythagorean Theorem, or
from the Distance Formula:
The direction of a vector is determined by the angle it makes with a
horizontal line. In the diagram at the right, to find the direction of the
vector (in degrees) we will utilize trigonometry. The tangent of the
angle formed by the vector and the horizontal line (the one drawn
parallel to the x-axis) is 4/6 (opposite/adjacent).
Scalar (dot) product of vectors.
One can formally define an operation of scalar multiplication on
vectors, consistent with the following definition of length, or
magnitude of a vector,
.
(8)
and the following properties, which hold if , , and
and r is a scalar.
are real vectors
C
 The dot product is commutative:
A
B
 The dot product is distributive over vector addition:
 The dot product is bilinear:
 When multiplied by a scalar value, dot product satisfies:
(these last two properties follow from the first two). Then, it is clear
from the drawing at the right, that
.
(9)
Two non-zero vectors a and b are orthogonal if and only if a • b = 0.
Unlike multiplication of ordinary numbers, where if ab = ac, then b
always equals c unless a is zero, the dot product does not obey the
cancellation law: if a • b = a • c and a ≠ 0, then we can write: a • (b − c)
= 0 by the distributive law; the result above says this just means that
a is perpendicular to (b − c), which still allows (b − c) ≠ 0, and therefore
b ≠ c.
The coordinate representation for the dot product is (a∙b) = axbx + ayby.
According to the definition given above, this must be equivalent to
(a∙b) = ab cos (
), which could be straightforwardly verified from
the drawing.
Definition of ellipse, hyperbola and parabola: Tangent circles.
Ellipse is the locus of centers of all circles
tangent to two given nested circles (F1,R)
and (F2,r). Its foci are the centers of these
given circles, F1 and F2, and the major axis
equals the sum of the radii of the two
circles, 2a = R+r (if circles are externally
tangential to both given circles, as shown in
the figure), or the difference of their radii
(if circles contain smaller circle (F2,r)).
Consider circles (F1,R) and (F2,r) that are
not nested. Then the loci of the centers O
of circles externally tangent to these two
satisfy |OF1| - |O F2| = R - r.
Y
O
r’
r’
R
F2
F1
0
r
X
Y
F2
F1
0
r
X
R
r’
Hyperbola is the locus of the centers of
O
circles tangent to two given non-nested
circles. Its foci are the centers of these
given circles, and the vertex distance 2a equals the difference in radii
of the two circles.
r’
As a special case, one given circle may be a point located at one focus;
since a point may be considered as a circle of zero radius, the other
given circle—which is centered on the other focus—must have radius
2a. This provides a simple technique for constructing a hyperbola. It
follows from this definition that a tangent line to the hyperbola at a
point P bisects the angle formed with the two foci, i.e., the angle F1PF2.
Consequently, the feet of perpendiculars drawn from each focus to
such a tangent line lies on a circle of radius a that is centered on the
hyperbola's own center.
If the radius of one of the given circles is zero, then it shrinks to a
point, and if the radius of the other given circle becomes infinitely
large, then the “circle” becomes just a straight line.
Parabola is the locus of the centers of circles passing through a given
point and tangent to a given line. The point is the focus of the parabola,
and the line is the directrix.
Optical property of ellipse, parabla and hyperbola (continued).
If a light source is placed at one focus of an
elliptic mirror, all light rays on the plane of the
ellipse are reflected to the second focus. Since
no other smooth curve has such a property, it
can be used as an alternative definition of an
ellipse. (In the special case of a circle with a
source at its center all light would be reflected
back to the center.) If the ellipse is rotated
along its major axis to produce an ellipsoidal mirror (specifically, a
prolate spheroid), this property will hold for all rays out of the source.
Alternatively, a cylindrical mirror with elliptical cross-section can be
used to focus light from a linear fluorescent
lamp along a line of the paper; such mirrors are
used in some document scanners. 3D elliptical
mirrors are used in the floating zone furnaces
to obtain locally high temperature needed for
melting of the material for the crystal growth.
Sound waves are reflected in a similar way, so in
a large elliptical room a person standing at one
focus can hear a person standing at the other focus remarkably well.
In the 17th century, Johannes Kepler discovered that the orbits along
which the planets travel around the Sun are ellipses with the Sun at
one focus, in his first law of planetary motion.
The method of coordinates. Ellipse.
An ellipse is a smooth closed curve which is
symmetric about its horizontal and vertical axes.
The distance between antipodal points on the
ellipse, or pairs of points whose midpoint is at the
center of the ellipse, is maximum along the major
axis or transverse diameter, and a minimum along
the perpendicular minor axis or conjugate
diameter.The semi-major axis (denoted by a in
the figure) and the semi-minor axis (denoted by b in the figure) are
one half of the major and minor axes, respectively. These are
sometimes called (especially in technical fields) the major and minor
semi-axes, the major and minor semiaxes, or major radius and minor
radius.
The foci of the ellipse are two
special points F1 and F2 on the
ellipse's major axis and are
equidistant from the center point.
The sum of the distances from
any point P on the ellipse to those
two foci is constant and equal to
the major axis ( PF1 + PF2 = 2a ).
Each of these two points is called
a focus of the ellipse.
Excersise. Prove that the sum of the distances from any point inside
the ellipse to the foci is less — and from any point outside the ellipse is
greater — than the length of the major axis.
Consider the locus of points such that the sum of the distances to two
given points A(-f,0) and B(f,0) is the same for all points P(x,y). It is an
ellipse with the foci A and B. This distance is equal to the length of the
major axis of the ellipse, 2a. Then, for every point on the ellipse,
,
Where from, by squaring this equation twice, we obtain for an ellipse,
.
The equation of an ellipse whose major and minor axes coincide with
the Cartesian axes is
. The area enclosed by an ellipse is πab,
where (as before) a and b are one-half of the ellipse's major and minor
axes respectively.
The eccentricity of an ellipse, usually denoted by ε or e, is the ratio of
the distance between the two foci, to the length of the major axis or e
= 2f/2a = f/a. For an ellipse the eccentricity is between 0 and 1 (0 < e <
1). When it is 0 the foci coincide with the center point and the figure
is a circle. As the eccentricity tends toward 1, the ellipse gets a more
elongated shape. It tends towards a line segment if the two foci
remain a finite distance apart and a parabola if one focus is kept fixed
as the other is allowed to move arbitrarily far away. The distance ae
from a focal point to the centre is called the linear eccentricity of the
ellipse (f = ae).
Other definitions of the ellipse.
Each focus F of the ellipse is associated with a line parallel to the
minor axis called a directrix. Refer to the figure. The distance from
any point P on the ellipse to the focus F is a constant fraction of that
point's perpendicular distance to the directrix resulting in the equality,
e =PF/PD < 1.
In stereometry, an ellipse is defined as a plane curve
that results from the intersection of a cone by a
plane in a way that produces a closed curve. Circles
are special cases of ellipses, obtained when the
cutting plane is orthogonal to the cone's axis.
Consider a circle with the center O, which is
internally tangent to two given circles with
centers F1 and F2 and radii R and r,
respectively, such that one is inside the other
(see figure). The sum of the distances from
O to the centers F1 and F2, equals |OF1| + |O
F2| = R + r, and is independent of the position
of the circle (O,r’). Therefore, the locus of
centers of all such circles [internally tangent
to two given nested circles (F1,R) and (F2,r)] is
an ellipse with foci F1 and F2.
Consider circles (F1,R) and (F2,r) that are not
nested. Then the loci of the centers O of
circles externally tangent to these two
satisfy |OF1| - |O F2| = R - r.
Y
r’
r’
O
R
r
F2
F1
0
X
Y
F2
F1
0
r
X
R
r’
r’
O
Y
The method of coordinates. Parabola.
Consider locus of points equidistant from a
given point F and a given line l, which does not
contain this point. It is a parabola with focus F
and the directrix l. The line perpendicular to
the directrix and passing through the focus
(that is, the line that splits the parabola
through the middle) is called the "axis of
y = kx2
(0,a)
X
O
(0,-a)
y = -a
symmetry". The point on the axis of symmetry that intersects the
parabola is called the "vertex", and it is the point where the curvature
is greatest.
The easiest way to show that the above definition indeed corresponds
to a parabola, is by using the method of coordinates. Indeed, let line l
be parallel to the X-axis and intersect the Y-axis at y = -a, and focus F
(0, a) lie on the Y-axis at the same distance a from the origin. Then for
any point on the parabola according to the
definition above,
, wherefrom
Y
.
If the directrix is a line y = a, and the focus has
some general coordinates, F(xF, yF), then the
points on the parabola satisfy the equation
, where
and
y = k(x-xF)2+b
(xF,yF)
. If the directrix is parallel to the Y-
y=a
axis, then parabola’s equation becomes x = k (y yF)2+b.
Parabolas can open up, down, left, right, or in
some other arbitrary direction. Any parabola
can be repositioned and rescaled to fit exactly
on any other parabola — that is, all parabolas
are similar. How do you think the equation of a
parabola with the directrix y = ax + b, at an
arbitrary angle atan(a) to the (X,Y) axes, looks
like? In order to solve this problem, we need to
learn how to find a distance from the point
P(x,y) to a line y = ax + b.
(0,a)
X
O
Y
(xF,yF)
y=
ax
+b
(0,b)
O (-B/a,0)
X
Alternatively (in stereometry), parabola is
defined as a conic section, similar to the
ellipse and hyperbola. Parabola is a unique
conic section, created from the intersection
of a right circular conical surface and a plane
parallel to a generating straight line of that
surface. The parabola has many important
applications, from automobile headlight
reflectors to the design of ballistic missiles.
They are frequently used in physics,
engineering, and many other areas.
The method of coordinates. Hyperbola.
Hyperbola can be defined as the locus of
points where the absolute value of the
difference of the distances to the two foci is
a constant equal to 2a, the distance between
its two vertices. This definition accounts for
many of the hyperbola's applications, such as
trilateration; this is the problem of
determining position from the difference in
arrival times of synchronized signals, as in
GPS.
Similarly to the case of an ellipse, we write,
,
where (f,0) and (-f,0) are the positions of the
two foci, which we chose to lie on the X-axis.
Squaring the above equation twice, we arrive
at the canonical equation for the parabola,
The asymptotes of the hyperbola
(red curves) are shown as blue
dashed lines and intersect at the
center of the hyperbola, C. The two
focal points are labeled F1 and F2,
and the thin black line joining them
is the transverse axis. The
perpendicular thin black line
through the center is the
conjugate axis. The two thick black
lines parallel to the conjugate axis
(thus, perpendicular to the
transverse axis) are the two
directrices, D1 and D2. The
eccentricity e equals the ratio of
the distances from a point P on the
hyperbola to one focus and its
corresponding directrix line (shown
in green). The two vertices are
located on the transverse axis at
±a relative to the center. So the
parameters are: a — distance from
center C to either vertex;
b — length of a perpendicular
segment from each vertex to the
asymptotes; c — distance from
center C to either Focus point, F1
and F2, and θ — angle formed by
each asymptote with the
transverse axis.
, or,
, where
.
A hyperbola consists of two disconnected curves called its arms or
branches. At large distances from the center, the hyperbola
approaches two lines, its asymptotes, which intersect at the
hyperbola's center. A hyperbola approaches its asymptotes arbitrarily
closely as the distance from its center increases, but it never
intersects them; however, a degenerate hyperbola consists only of its
asymptotes. Consistent with the symmetry of the hyperbola, if the
transverse axis is aligned with the x-axis of
a Cartesian coordinate system, the slopes of
the asymptotes are equal in magnitude but
opposite in sign, ±b⁄a, where b=a×tan(θ) and
where θ is the angle between the transverse
axis and either asymptote. The distance b
(not shown) is the length of the
perpendicular segment from either vertex to
the asymptotes.
A conjugate axis of length 2b, corresponding
to the minor axis of an ellipse, is sometimes
drawn on the non-transverse principal axis;
its endpoints ±b lie on the minor axis at the height of the asymptotes
over/under the hyperbola's vertices. Because of the minus sign in some
of the formulas below, it is also called the imaginary axis of the
hyperbola.
If b = a, the angle 2θ between the asymptotes equals 90° and the
hyperbola is said to be rectangular or equilateral. In this special case,
the rectangle joining the four points on the asymptotes directly above
and below the vertices is a square, since the lengths of its sides 2a =
2b.
If b = a, the angle 2θ between the asymptotes equals 90° and the
hyperbola is said to be rectangular or equilateral. In this special case,
the rectangle joining the four points on the asymptotes directly above
and below the vertices is a square, since the lengths of its sides 2a =
2b.
If the transverse axis of any hyperbola is aligned with the x-axis of a
Cartesian coordinate system and is centered on the origin, the equation
of the hyperbola can be written as
A hyperbola aligned in this way is called an "East-West opening
hyperbola". Likewise, a hyperbola with its transverse axis aligned with
the y-axis is called a "North-South opening hyperbola" and has equation
Every hyperbola is congruent to the origin-centered East-West opening
hyperbola sharing its same eccentricity ε (its shape, or degree of
"spread"), and is also congruent to the origin-centered North-South
opening hyperbola with identical eccentricity ε — that is, it can be
rotated so that it opens in the desired direction and can be translated
(rigidly moved in the plane) so that it is centered at the origin. For
convenience, hyperbolas are usually analyzed in terms of their centered
East-West opening form.
Other definitions of the hyperbola.
The above definition may also be expressed in terms of tangent circles.
The center of any circles externally tangent to two given circles lies on
a hyperbola, whose foci are the centers of the given circles and where
the vertex distance 2a equals the difference in radii of the two
circles. As a special case, one given circle may be a point located at one
focus; since a point may be considered as a circle of zero radius, the
other given circle—which is centered on the other focus—must have
radius 2a. This provides a simple technique for constructing a
hyperbola. It follows from this definition that a tangent line to the
hyperbola at a point P bisects the angle formed with the two foci, i.e.,
the angle F1P F2. Consequently, the feet of perpendiculars drawn from
each focus to such a tangent line lies on a circle of radius a that is
centered on the hyperbola's own center.
In stereometry, hyperbola can be defined
as a conic cross-section, similar to parabola
and ellipse. Namely, hyperbola is the curve
of intersection between a right circular
conical surface and a plane that cuts
through both halves of the cone. For the
other major types of conic sections, the
ellipse and the parabola, the plane cuts through only one half of the
double cone. If the plane is parallel to the axis of the double cone and
passes through its central apex, a degenerate hyperbola results that is
simply two straight lines that cross at the apex point.
Directrix and focus
Similarly to an ellipse, a hyperbola can be defined as the locus of points
for which the ratio of the distances to one focus and to a line (called
the directrix) is a constant ε. However, for a hyperbola it is larger
than 1. This constant is the eccentricity of the hyperbola. By symmetry
a hyperbola has two directrices, which are parallel to the conjugate
axis and are between it and the tangent to the hyperbola at a vertex.
Reciprocation of a circle*
The reciprocation of a circle B in a circle C always yields a conic section such as a
hyperbola. The process of "reciprocation in a circle C" consists of replacing every
line and point in a geometrical figure with their corresponding pole and polar,
respectively. The pole of a line is the inversion of its closest point to the circle C,
whereas the polar of a point is the converse, namely, a line whose closest point to C
is the inversion of the point.
The eccentricity of the conic section obtained by reciprocation is the ratio of the
distances between the two circles' centers to the radius r of reciprocation circle
C. If B and C represent the points at the centers of the corresponding circles,
then
Since the eccentricity of a hyperbola is always greater than one, the center B must
lie outside of the reciprocating circle C.
This definition implies that the hyperbola is both the locus of the poles of the
tangent lines to the circle B, as well as the envelope of the polar lines of the points
on B. Conversely, the circle B is the envelope of polars of points on the hyperbola,
and the locus of poles of tangent lines to the hyperbola. Two tangent lines to B
have no (finite) poles because they pass through the center C of the reciprocation
circle C; the polars of the corresponding tangent points on B are the asymptotes of
the hyperbola. The two branches of the hyperbola correspond to the two parts of
the circle B that are separated by these tangent points.
Review the following problems from recent homework.
1.
A and B are given points, k is a given constant, SMAB is the area of
triangle MAB. Prove that the locus of points M, such that
, is a pair of straight lines.
2.
Find the equation of the locus of points equidistant from two
lines,
and
, where a, b, m, n are real
numbers.