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Conics reprezent a class of plain curves with special properites, which have numerous applications in various domains.
The mathematicians of the elenic era obtained nondegenerated conics by the intersection of a rotation conic with a plane. The books of Apollonius (262200 BC) describe for the first time the elypse, hyperbola, parabola and their properties.
We will make a short introduction into the world of conics by using the reduced equation, then we will classify izometracally the curves of 2nd order. Using the knowledge from the previous chapters, we will understand that the geometrical locus of points in the euclidian plane, in an othonormed frame is the intersection of a circular cone with a plane, or Null.
The general properties of conicsm and the conics with initial conditions, will be studied in the final part of this chapter.
Let be E_{ } the bidimensional punctual euclidian space, and
R (O,_{}) a carthesian orthonormed frame. A plane curve is determined in a carthesian refrence frame by an equation with 2 variables: F ( x,y ) = 0. A particular case of the plane curves are the conics. Behold, the geometrical definitions and the algebrical properties of conics in a freely chosen refrence frame:
1.1 Elypse: The geometrical locus of points belonging to the euclidian plane with the property that the sum of distances from 2 fixed points F_{1} and F_{2} stays constant.
Given xOy is an orthogonal reference frame in the euclidian plane E_{2} aI R and the points F_{1 (c,0) }, F_{2 (c,0)} are fixed, then the all points M_{(x,y)} I E_{2} with the property MF_{1} + MF_{2} = 2a are algebrically written as follows:
_{} , c = _{} (1.1)
y
B M_{(x,y)}
_{}
A Ooo A x
x =  _{} B’ x = _{}
fig.1
For the elypse (1.1) following notions are to be considered: (fig.1) :
The Axes Ox ,Oy of the carthesian frame are axes of symmetry, the origin of the frame is the center of the elypse. Therefore the orthonormed frame R(O,_{} ) is also called canonical and the equation (1.1) is called reduced.
The elypse (1.1) is the geometrical locus of the points M_{(x,y)} with satisfy one of the following properties:
_{} or _{}
The elypse of with the semiaxis a and b respectively, can be parametrically written as: x = a cosj , y = b sin j , j I p] .
It can be easily proven that the perpendicular on the tangent of any point of the elypse is the bisecting line of the angle between the focal radiuses to that point. (The optical property of the elypse).
1.2 Hyperbola : The geometrical locus of the points from the euclidian plane E_{2} for which the modulus of the diffrence of distances of two distinct, fixed points is constant.
Given xOy an orthogonal refrence frame in the euclidian plane E_{2}, aI R and the fixed points F_{1 (c,0) }, F_{2 (c,0)} then the points M_{(x,y)} I E_{2} with the property MF_{1}  MF_{2} = 2a is algebrically written as:
_{} , c = _{} (1.2)
y
F_{1} A^{’} O A F_{2}
Fig.2
The axes Ox , Oy of the frame R(O,_{} ) are the symmetry axes of the hyperbola, and the origin of the frame is a symmetry center of the hyperbola, hence equation (1.2) is the reduced form of the hyperbola.
The Hyperbola characterized by the equation (1.2) is the geometrical locus of the points M_{(x,y) }I E_{2} , which satisfy one of the following conditions:
_{} or _{},
where the lines d_{1 } and d_{2 } are the directing lines of the hyperbola.
The parametric equations of the hyperbola are given by:
x = a ch t , y = b sh t , tI R
The tangent to a point of the hyperbola, is the bisecting line of the angle between the focal radiuses. ( the optical property of the hyperbola )
1.3 Parabola : the geometric locus of points equally distanced from a fixed point F (focal) and a fixed line D (directing line) .
If xOy is an orthogonal refrence frame in the euclidian plane E_{2} , pI R _{+} , and given the point F_{ (,0) }and the line (D): x =  _{} then the set of points M_{(x,y)} withthe property _{}(M,F) = _{}(M, D) is algebrically written as follows:
y^{2} = 2px , ( p > (1.3)
D y
M_{(x,y)}
O F(_{},0) x
Fig.3
We define the following notions associated to a parabola:
F(_{},0) – focal point of the parabola, the quantity _{} focal distance
O(0,0)  edge of the parabola
Ox  transversal axis of the parabola, the symmetry axis
Oy  tangent axis of the parabola
the line D of equation x =  _{} , is the directing line
the excentricity of the parabola is e=1.
In the euclidian punctual space, referred to an affine reference frame the polinomial function (affine form) f :R^{2} R , is given by:
f (x,y) = a_{ }x^{2} + 2a_{ }xy + a_{ }y^{2} + 2a_{ }x + 2a_{ }y + a_{ } , a_{ }^{ } a_{ }^{ } a_{ }^{ }
Finally we will proove, that the set of points of the plane E_{2} , whose coordinates (x,y) are the roots of the function f, represent a conic, or a Null space, from the geometrical point of view, argumenting the following definition:
Definition. 2.1 
The algebrical curve of second order, which includes the set of points of the plane E_{2} is called a conic, if it’s coordinates satisfy the following equation: 
a_{ }x^{2} + 2a_{ }xy + a_{ }y^{2} + 2a_{ }x + 2a_{ }y + a_{ } = 0 , a_{ }^{ } a_{ }^{ } a_{ }^{ } (2.1)
If we consider the homogenous coordinates (x_{1},x_{2},x_{3}) of a point M_{(x,y)} , tied to the coordinates (x,y), called nonhomogenous coordinates, trough the relations: x = _{} , y = _{}, we obtain:
f(x_{1},x_{2},x_{3}) s a_{ }x_{ }_{ }^{ } + 2a_{ }x_{ }x_{ } + a_{ }x_{ }^{ } + 2a_{ }x_{ }x_{ } + 2a_{ }x_{ }x_{ } + a_{ }x_{ }^{ }
or _{} , where a_{ij} = a_{ji } _{ } (2.2)
The symmetrical matrix A = (a_{ij}) will be called the matrix of the conic (_{}).
Let us consider the numbers:
D = det.A , _{} , I = a_{11} + a_{22 }.
If we apply a (izometrical) orthogonal transformation T : V_{ } V_{ } , T (_{}) = _{}, T(_{})=_{}, in the euclidian punctual space E_{ }, passing from the orthonormed frame R (O,_{}) to the new orthonormed frame
R (O,_{}) – the conic (_{}) will have the following analytic equation:
_{}, where D d , I are calculated in the new frame.
Theorem 2.2 
The quatities D d , I are the orthogonal invariants of the conic ( g ), hence : D D, d d , I = I . 
These invatriants will be called as follows: Dcubic invariant,
d  square invariant, I – linear invariant .
Proof. We will demonstrate first the invariance of d and I. Let us consider the quadratic form:
j (x,y) = a_{ }x^{2} + 2a_{ }xy + a_{ }y^{2} (2.3)
The symmetrical bilinear form, associated to the quadratic form j, has a unique symmetrical linear transformation T : E_{2 } E_{2 }having its associated matrix identical to that of the quadratic form (2.3). Hence the characteristic equation of the transformation T
_{} l^{ } – I l d = 0,
is invariant to a base change, therefore d d , I = I .
Writing the equation of the conic (_{}) with the homogenous coordinates (2.2) and repeating the procedure described above for the quadratic form f(x_{1},x_{2},x_{3}), we can proove the invariance of the characteristic equation:
_{} l^{ } J_{1}l^{ } + J_{2}l D
therefore D is also invariant.
Definition 2.3 
The symmetry center of the set of points of the conic (_{}), is called the center of the conic (_{}). 
If the conic (_{}) would be written analytically as the following equation: a_{ }x^{2} + 2a_{ }xy + a_{ }y^{2}+ k = 0, then the origin of the frame would be the symmetry center of the conic, f(x,y) = f (x,y). Let us determine de conditions where the conic (_{}) admits a center, if so, we’ll also try to find it.
Using the translation from frame R (O,_{}) to the point C(x_{o},y_{o}):
_{} (2.4)
the equation (2.1) will be written under the following form:
a_{ }x’^{2} + 2a_{ }x’y’ + a_{ }y’^{2} + 2(a_{ }x_{o} + a_{ }y_{o} + a_{ })x’ +
+ 2(a_{ }x_{o} + a_{ }y_{o} + a_{ } )y’ + f(x_{o},y_{o}) = 0
Applying the symmetry condition f(x’,y’) = f(x’,y’) we get:
_{} or _{} (2.5)
The equations (2.5) represent the equations of the center, if it exists.
The following cases are to be considered:
a) _{}, system (2.5) has a unique solution, the point C(x_{o},y_{o}) is the center of the conic (_{}).
b) _{}, system (2.5) has no solution, or admits an infinity of solutions, which means that the conic (_{}) has no unique center given a finite distance.
In order to recognize what the equation (2.1) geometrically represents, we will determine the frame in which the equation has its simplest form using an izometric transformation. This form will be called the canonical form of the conic (_{}). This way, we have proven that the equation (2.1) is equivalent to one of the reduced equations described above, or with the empty set. The problem will be handled depending on the value of d which is either d 0 or d
2.1 Reducing to the canonical form of conics with center , d
Consider a conic (_{}) analytically represented by the equation (2.1) with the center in point C(x_{o},yo). Translating the frame R (O,_{}) to the point C(x_{o},y_{o}) , (equations (2.4)) we obtain:
a_{ }x’^{2} + 2a_{ }x’y’ + a_{ }y’^{2} + k = 0 , k = f(x_{o},y_{o}) (2.6)
Let’s consider the quadratic form _{} a_{ }x’^{2} + 2a_{ }x’y’ + a_{ }y’^{2} and the assosicated transformation T : E_{2} E_{2} having the same matrix as the quadratic form j We know that an orthonormed frame exists, consisting of the eigenvectors _{ }of the transformation T, hence we can write the quadratic from j as a sum of squares :
_{}l_{ }X^{2} + l_{ } Y^{2}, (2.7)
where l_{ } and l_{ } are the eigenvalues of the transformation T , solutions of the characteristic equation:
_{} or l^{ } –I l d (2.8)
Equation (2.8) is called secular equation, and it always has a solution because the matrix (a_{ij}), i,j=1,2 is symmetrical.
Both frames are orthonormed, therefore the translation from
R (C,_{}) with axes Cx and Cy to the frame R (C,_{}) with axes CX and CY is obtained using an izometrical transformation trough C as a fixed point.
Noting with (x_{ } x_{ }) and (h_{ } h_{ }) the coordinates of the eigenvectors _{}and _{} in the frame R (C,_{}) , the coordinate transformation is given by
_{} , (2.9)
where the translation matrix is R = _{} and is orthogonal, and has det.R = 1, meaning that the traslation from R to R is done by using a curl when det.R = 1, followed by a symmetry if det.R =  1 .
In the new frame determined by the eigenvectors _{}and _{}, having the center of the conic as an origin, the equation of the conic (2.1) becomes:
l_{ }X^{2} + l_{ } Y^{2} + k = 0 (2.10)
If we use the invariance of D at orthogonal transformations, and we calculate it for the carthesian frame XCY, we obtain D = k l_{ }l_{ } = k d ( d l_{ }l_{ } from the secular equation) therefore, k = _{} . Hence, in the frame R (C,_{}) the equation of the conic (_{}) is written as:
(_{}): l_{ }X^{2} + l_{ } Y^{2} + _{} = 0 (2.11)
and is called the canonical form of the equation of the conic.
In the case of conics with center, the canonical equation (2.11) can be written easily if the orthogonal invariants D d and I are known.
2. When reducing to the canonical from the equation of the conic with center, the order of the transformations is not essential: first the translation to the center of the conic and then the fixed point isometry, or viceversa.
3. The coordinate axes CX and CY are symmetry axes for the conic (g), therefore the frame R (C,_{}) is the frame where the conic can be written under a reduced form, hence it can be recognized geometrically.
The slopes of these symmetry axes can be determined using the coordinate expression of the eigenvectors. If the eigenvectors _{}and _{} are normed, corresponding to the eigenvalues l_{ }and l_{ }then the frame
R (C,_{}) is obtained through an angle rotation where aI p], from frame R (O,_{}) if and only if for _{}= (x_{ } x_{ }) we have _{}. Hence, we can write the equations which determine these eigenvectors:
_{} and _{}
noting m = tg a =_{} we obtain
_{} , which is equivalent to
a_{11} m^{2} + (a_{11} – a_{22 }) m – a_{12} = 0 , (2.12)
called the slope equation of the symmetry centers of a nondegenerated conic with the center at a finite distance. The equations of the axes of the conic, can be written as the line equations which pass through point C(x_{o},y_{o}) and have the slopes m_{1} and m_{2} , the solutions of equation (2.12) .
To find the canonical frame using a translation followed by a rotation (curl) we will choose the order (and signs) if the eigenvectors accordingly, so that det.R = 1.
In order to determine the rotation angle, when passing from frame R (C,_{}) to frame R (C,_{}), we will write the equations which determine the eigenvectors:
_{} and _{} . (2.13)
Denoting the angle between _{} with q I (0,_{}) , we choose the sign of _{} so that the agle between _{} and _{} to be q, hence_{}
_{} and _{}
From relations (2.12) we get :
_{} and _{} , therefore
_{} , hence
tg 2q =_{} (2.14)
The same relations show that : _{} , hence
sign l_{ } l_{ }) = sign (a_{ }
5. By considering the analytical characteristics of the izometrical transformations, the reduction to the canonical form of a conic can be done using the rototranslation method. We operate the rotation
_{} (2.15)
and we determine the angle j where the coefficients of x y must be zero. After fulfilling this conditions and grouping the terms as a sum of squares, we can do the translation to the center of the conic.
If we wish to draw the conic (g) the following steps have to be made:
 we write the conic (g) in the canonical form (2.11)
 we plot the center C(x_{o},y_{o}) with respect to the xOy carthesian reference frame, then the center and the axes of the canonical reference frame CX and CY (with direction and sign), determined from the eigenvectors _{} and _{} plotted in point C.
 in the carthesian frame XCY we draw the conic (g) according to the canonical form (2.11)
Returning to the canonical form (2.11), lets analyze the following situations:
Case 1^{o} D
a) d > T l_{ }l_{ } d > 0 , equation (2.11) can be written in one of the following forms:
_{} or _{} , (2.16)
hence, we have the case of a real elypse, or the case of the empty set.
b) d < T l_{ }l_{ } d < 0 , equation (2.11) can be written in one of the following forms:
_{} or _{}, (2.17)
hence the conic (g) represents a hyperbola. If I = a_{ } + a_{ } l_{ } l_{ }
that would mean a = b and the hyperbola (2.16) is equilateral.
Case 2^{o} D
a) d > T l_{ }l_{ } d > 0 , the equation is written under the form
a^{ }X^{2} + b^{ }Y^{2} = 0 .
In such case, the conic is reduced to a point, the center C(x_{o},y_{o}) .
b) d < T l_{ }l_{ } d < 0 , equation (2.11) can be written in the form:
a^{ }X^{2}  b^{ }Y^{2} = 0 (a^{ }X  bY) (a^{ }X + bY) = 0 , (2.19)
hence the conic is in fact two nonparallel lines.
As a conclusion, we can state that the cubic invariant D gives us information about the nature of the conic, and the square invariant d gives us information about the type of the conic (g). Therefore we can say:
if D  we have a nondegenerated conic
g) : a_{ }x^{2} + 2a_{ }xy + a_{ }y^{2} + 2a_{ }x + 2a_{ }y + a_{ } = 0 , a_{ }^{ } a_{ }^{ } a_{ }^{ }
and the line which passes in the point M(x_{o},y_{o}) and the direction given by _{} = (l,h)
(d) : x = x_{o} + lt , y = y_{o} = ht , tI R .
In order to study thre relative position of the line (d) with respect to the conic (g) we must study the set of solutions of the system formed by the equations of the line, and equation of the conic. If we enter the coordinates of a point belonging to the line into the equation of the conic, we obtain the equation of second order in the variable t I R :
j(l,h) t^{ } l _{} + h _{} t + _{} = 0 (2.32)
where:
_{}
_{} (2.33)
_{}
It is clear that a line intersects a conic in maximum two points.
We have the following cases:
Case 1^{o} If j(l,h) 0 , for
a) l _{} + h _{} j(l,h)_{} > 0, the equation (2.32) has two real solutions t_{1} t_{2} and the line interesects the conic in 2 points M_{1} M_{2}.
b) l _{} + h _{})^{2}  j(l,h)_{} = 0, the equation (2.32) has two identic real solutions t_{1}=t_{2,} and the line interesects the conic in two identic points M_{1}=M_{2, }hence the line is tangent to the conic in point M_{1}.
Definition.2.4 
The direction _{} with the property _{} (2.35) is called asymptotic direction of the conic (g 
Definition.2.5 
A line that does not intersect the conic, and has asymptotical direction is called the asymptote of the nondegenerated conic g 
Realizantul of the equation (2.35), with the variable m = _{}, is given by (a_{12})^{2} a_{11}a_{22 }=  d. Therefore, if
d > g) is an elypse and has no asymptotical directions
d g) is a parabola and admits a double asymptotical direction which is the direction of the parabola axis.
d < g) is a hyperbola, and admits 2 asymptotical directions, and the lines of these directions pass through the center of the conic, and are the aymptotes of the hyperbola given by this equation:
a_{22} m^{2} + 2a_{12} m + a_{11} = 0 (2.36)
The Diameter of conic conjugated with a given line.
Consider the conic (g) given by the equation (2.1) and a fixed direction given by the vector _{}. The lines of the line family parallel to the direction (l,h), intersect the conic (g) in maximum 2 points. If M_{1} and M_{2} are the intersection points of the conic with a line of direction _{} , then we have the following theorem:
Theorem 2.6 
The geometrical locus of the middles of the segment M_{1}M_{2} given by the secant lines of the conic g ), with the direction _{}are a subset of a line. 
Proof. Consider the line (d) passing trough point M_{o}(x_{o},y_{o}), with nonasymptotic direction _{}
(d) : _{} ,
intersecting the conic in the points M_{1}(t_{1}) and M_{2}(t_{2}). The values t_{1} and t_{2 } corresponding to the intersection points M_{1} and M_{2} are the solutions of the equation (2.23). Without endangering the generality, let us consider the point M_{o}Id the middle of the segment M_{1}M_{2} . The coordinates if the point M_{o} are
_{} and _{} , therefore,
t_{1} + t_{2} = 0 , and _{} .
If M_{o} is the middle of the arbitrary cord of direction _{}, the coordinates of the middles of the segment M_{1}M_{2} satisfy the equation:
_{} (2.37)
or
_{}, (2.37)’
practically, a line. Q.e.d.
Definition.2.7 
The line conjugate to the nonasymptotic direction _{}and given by the equation _{} (2.38) is called the diameter of the conic (g 
If d 0 and _{} arbitrary, the equation (2.38) represents the pencil of planes passing through the center of the conic (g), therefore the conjugate of a given direction is a line passing through the center of the conic.
If d = 0 and D 0, the equation (2.38) has the form f_{x}+ l = 0, lIR, which represents a pencil of parallel planes with the direction _{}, which is exactly the symmetry axis of the parabola. Therefore, the axis of the parabola is a conjugate of the diameter with the perpendicular direction to the axis, _{}.
The diameter conjugate to the direction m= _{}, equation (2.38) can be written in the following form:
_{} (2.39)
and has the slope m given by
_{} , or (2.40)
_{} (2.40)’
called relatia de conjugare .
Two lines with nonasymptotical directions, passing through the center of the conic (g d 0) and with their slopes m and m satisfy the equation (2.40) and are called the conjugate diametre of eachother.
Remarks:
1. The othogonal conjugate diametres define the symmetry axes of a conic with a center. Respecting the condition mm =  1, from equation (2.40), we obtain the equations of the slopes of the symmetry axes of the conic (g
_{},
hence, the same with equation (2.12) .
2. Selfconjugate diametres, m = m, define the asymptotes of the conic (g) From the equation (2.40) we obtain the equations of the slopes of the conic asymptotes (hyperbola)
_{}, the same with equation (2.36) .
Let us consider the fixed point M_{o}(x_{o},y_{o}) and (d), a line with variable direction _{} passing through point M_{o}: x=x_{o}+lt, y=y_{o}+ht.
Consider now the points M_{o }, M_{1}, M_{2 } and M on the line (d) , characterized by the parametrical coordinates t_{o}, t_{1}, t_{2}_{ } and t_{ }.
Definition.2.8 
The point M is called the armonical conjugate of the point M_{o} with respect to M_{1 } and M_{2 } if _{} , (2.41) M_{i}M_{j } being an oriented segment. 
Relation (2.41) is written in parametrical coordinates like this:
_{} or _{} (2.42)
If the points M_{1 }and M_{2} are the intersection points of the line (d) with the conic(g) , then the following theorem is true:
Theorem 2.9 
The geometrical locus of point M , the armonical conjugate of point M_{o} with respect to points M_{1} and M_{2} , is a subset of the line with the following equation: 
_{},
(2.43)
Proof. Using the relations (2.42) and (2.32) we obtain
_{} (2.44)
By eliminating the parameters t, l , h from the equation (2.44) and the equations of the line (d) we obtain:
_{}, hence we’ll get equation (2.43)
Equation (2.43) can be obtained from the general equation of a conic (2.1) by using the following substitutions, called halvings:
_{} si _{}
Definition.2.10 
The line with the equation (2.43) is called the polar of the point M_{o} with respect to the conic g 
The point M_{o } who’s polar with respect to the conic g) is the line (d), is called the pole of the line (d).
Observation 3.
If point M_{o}(x_{o},y_{o}) belongs to the conic (g) , then f x_{o},y_{o}) = 0 considering that the direction of the tangent through point M_{o} is given by _{}, the equation _{} is equivalent to the equation (2.43) , hence, the equation of the tangent in the point M_{o} .
Conics defined by initial conditions.
Consider the conic (g) given by the general equation (2.1)
f (x,y) s a_{ }x^{2} + 2a_{ }xy + a_{ }y^{2} + 2a_{ }x + 2a_{ }y + a_{ } = 0, a_{ }^{ } a_{ }^{ } a_{ }^{ }
The six coefficients from the equation of the conic can’t be all zero at the same time, therefore this equation is equivalent with an equation which depends on only five coefficients, called esential parameters, obtained when dividing the initial six with one of them, nonzero. Therefore, in order to determine a conic uniquely, we need five conditions. For example a conic given by five points has the following equation:
_{} (2.45)
Consider the conics : (g_{ } f x,y) = 0 and (g_{ }) : g x,y
Definition.2.11 
The set of conics given by the general equationG _{}, _{} (2.46) is called pencil of conics given by the fundamental conics g_{ }) and (g_{ } 
The conics (g_{ }) and (g_{ }) belong to the pencil (G ) but equation (2.46) does not always represent a conic (case of simple points and lines). Knowing that a and b are not zero at the same time we can write the equation (2.46) in the following form:
_{}(G _{}, mIR ,
pencil, which does not contain the conic (g_{ }
If (g_{ }) and (g_{ }) intersect, then their intersection cannot contain more then four points. From earlier propsitions we conclude that through four noncollinear points pass an infinity of conics. If the distinct conics (g_{ }) and (g_{ }) are degenerate conics, degenerated into two lines, then the intersection points define in the plane a rectangle or a triangle.
Denoting with (AB) the left member of the equation of the line through A and B we have the following results:
a) The conics family circumscribed to the rectangle ABCD is given by the equation
a (AB) (CD) + b (BC) (AD) = 0 (2.47)
Particularly, the general equations of the conics which pass through the intersection of a conic f x,y) = 0 with two lines D_{ } D_{ } = 0, is given by:
a f x,y b D_{ }D_{ } (2.48)
If the line D = 0 intersects the conic f x,y) = 0 in two lines, we can consired it as the intersection between the degenerated conic D^{ } = 0 and the conic f x,y) = 0 , case in which the conics of the pencil (2.47) become bitangent to the conic f x,y) = 0 , in it’s intersection points with the line D=0 therefore, we have:
a f x,y b D^{ } (2.48)
In particular, the equation of the tangent conic to the lines D_{ } D_{ } = 0 in the points where these lines are intersected by the line D = 0 , hence they form a triangle, is given by:
a D_{ }D_{ } + b D^{ } (2.49)
b) Consider the conics (g_{ } f x,y g_{ } g x,y) = 0 and
g_{ } h x,y) = 0, then the set of conics given by the equation
a f x,y b g x,y m h x,y
is called the family of conics given by the conics (g_{ } g_{ }) and (g_{ }
Therefore the pencil of conics circumscribed to the triangle ABC is given by:
a (AB)(AC)+ b(BA)(BC) +m(CA)(CB) = 0 . (2.50)
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