Conic sections and reduction to metric canonical shape
Conic sections
1) Write the generic shape of one conical:
ax2 by2 cxy dx ey f = 0
2) When it can happen that the conic section is lacking in real points:
For the ellipses and the straight parallels when to 2° the member -1 appears, for the hyperbolas naturally cannot never happen.
3) As they are the relative autovalori you to 2 straight parallels:
One of the 2 autovalori is 0 and goes put which coefficient of y2 , dopodichè to continuation of the substitutions must be able to cancel also the coefficient of y.
Ellipse
4) Definition of ellipse:
It is the place of the equidistant points from 2 fixed points sayings fires.
5) Equation standard:
6) Fires:
if the focale axis is on the abscissas 
while if is on the formers 
.
7) Center of symmetry :
The point is the origin that is of coordinated (0,0).
8) In that point the ellipse meets the aces coordinate to you:
It meets the abscissas in the 
points
and the formers in the points 
9) Which it is the greater axle shaft and that meant it has for the ellipse:
It is the axle shaft correspondent to the greater one in absolute value between to and b.
10) optical Property of the ellipse:
The straight tangent to the ellipse in a data point forms equal angles with the focali beams.
11) Property of the autovalori of the Ellipse:
They are both positi to you.
12) Principle of assignment of the autovettori:
It convene to put smallest like coefficient than x2, in such a way the axis of the x becomes the axis focale.
13) When the conic section is reduced to a point:
When it has the equation of an ellipse but the famous term is 0.
14) When the conic section reduces to one circumference:
When it has the equation of an ellipse but the coefficients to and b they are both 1.
Hyperbola
15) Definition of hyperbola:
It is the place of the points for which the difference of the distances from 2 points of the plan said fires is constant.
16) Equation standard:
17) Fires:
The focale axis is the axis of the abscissas and the fires
have coordinated -
18) Center of symmetry :
It is origin (0,0).
19) Asymptotes:
They are the straight ones of equation 
.
20) You concern to us:
They are the points of coordinates .
21) Which it is the focale axis:
It is always the axis of the abscissas.
22) optical Property of the hyperbola:
The straight tangent to the hyperbola in a data point forms equal angles with the focali beams.
23) Property of the autovalori of the Hyperbola:
They are one positive and the other negative.
24) Principle of assignment of the autovettori:
If the famous term to 2° the member is negative to put the autovalore otherwise negative like coefficient ofx 2 viceversa. This could not be determining why the completion of the squares could alter the sign of the famous term.
25) When the conic section is reduced to 2 straight incidents:
When it has the equation of a hyperbola and the famous term are 0.
Parabola
26) Definition of parabola:
The distance from a fixed point of the said plan is the place of the points for everyone of which fire is equal to the straight distance from a fixed one, dictates director.
27) Equation standard:
y = ax2
28) Fire:
The fire has coordinated 
.
29) What is the straight director:
It is straight whose scope is described in the parabola
definition, it has equation 
30) Center of symmetry :
There is a symmetry axis, the axis of the formers.
31) optical Property of the parabola:
The straight tangent to the parabola in a data point forms equal angles with the focale beam and the semistraight parallel to the axis of outgoing symmetry from the bounce point.
32) Property of the autovalori of the parabola:
One of the 2 autovalori is 0.
33) Principle of assignment of the autovettori:
Which coefficient of y 2 convene to put autovalore0, so that the parabola turns the convexity towards the high.
Reduction to metric canonical Shape
34) Illustrate the steps of the reduction to metric canonical shape:
to) removing the mixed terms through an orthogonal transformation whose characteristics are gained from the autovalori
b) To remove the linear terms through a translation whose characteristics are gained from the completion of the squares.
c) To deduce the type of conic section and to characterize of the characteristic points.
d) To calculate the inverse transformations to the previous ones and to estimate the value of the characteristic points in the system of ccordinate originate them of the conic section.
f) To design the conic section.
35) That correspondence is between the autovalori in the diagonalizzata shape and the autovettori of the base ortonormalizzata in the matrix of change of base:
To the first autovalore of the diagonalizzata matrix the first carrier of the diagonalizzante orthogonal matrix corresponds.
36) That meant it has to diagonalizzare the quadratic part:
Spin or symmetry means to carry out an orthogonal transformation () that door the conic section in a system of typical reference of every conic section.
37) That meant it has to carry out the completion of the squares:
Translation of the conic section means to carry out one carrying some the center of symmetry in the origin.
38) That form it has the matrix of change of base from the base ortonormalizzata to the canonical base:
It is an orthogonal matrix having for columns the autovettori standardizes you of the quadratic shape.
39) That form it has the matrix of change of base from the canonical base to the ortonormalizzata base:
It is the inverse one of the matrix as soon as described and being it an orthogonal matrix, then the inverse one coincides with the transposed one.