## 385 Reputation

6 years, 291 days

## Reducing parabola with LinearAlgegra...

Maple 2021

how to transform this program using LinearAlgebra instead of linalg  ?

restart;
with(plots);
unprotect(gamma, D);
interface(rtablesize = 10): _EnvHorizontalName := 'x': _EnvVerticalName := 'y':
f := (x, y) -> 4*x^2 + 4*y*x + y^2 - 8*x + 16*y - 17: (for instance)
NULL;
Fg := proc(P::polynom, v::set, V::list, N::list)
local C, M, i, j;
C := coeffs(f(x, y), v, M); seq(`if`(member(op(i, [M]), N, 'j'), op(j, V) = op(i, [C]), NULL), i = 1 .. nops([M])); end proc:
Fg(f(x, y), {x, y}, [A, B, C, D, E, F], [x^2, y*x, y^2, x, y, 1]):
assign(%);
Delta := -4*A*C + B^2:
var := [x, y]:
with(linalg):
AA := matrix([seq([seq(diff(f(x, y), var[i], var[j])/2, j = 1 .. 2)], i = 1 .. 2)]):
vp := sort([eigenvals(AA)]):
print(`Valeur propres de AA ` = vp):
DD := jordan(AA, 'P11'):
print(`Matrice diagonale semblable à AA:   DD` = evalm(DD)):
G := map(normalize, GramSchmidt([col(P11, 1 .. 2)])):
PP := map(simplify, concat(op(G))):
print(`Matrice de passage orthogonale:   PP` = evalm(PP)):
print(`Directions principales de la conique:`):
print(`I1 ` = col(PP, 1), ` J ` = col(PP, 2)):
alpha := 1/2*arctan(B/(A - C)):
print('alpha' = evalf(%)):

M1 := matrix(1, 2, [X, Y]):
M2 := matrix(2, 1, [X, Y]):
multiply(M1, AA, M2):
N := matrix(1, 2, [D, E]):
multiply(M1, AA, M2) + multiply(N, M2):
multiply(M1, transpose(PP), AA, PP, M2) + multiply(N, PP, M2):

NULL;
The parabola in the new base is Y`^2 =X*(8*sqrt(5))/5 , <sqrt(5), -2*sqrt(5)>, <sqrt(5), 2*sqrt(5)>

Maple 2021

restart:
digits := 20;
unprotect(D);
G := 0.04361098108*x^2 + 0.4810001561*x*y + 1.326278064*y^2 - 0.7320831383*x - 2.656083763*y + 1 = 0
f := (x, y) -> lhs(G);
coeffs(f(x, y));
A, B, C, D, E, F := coeffs(f(x, y));
A := coeff(f(x, y), x, 2);
B := coeff(coeff(f(x, y), x), y);
C := coeff(f(x, y), y, 2);
D := coeff(coeff(f(x, y), x, 1), y, 0);
E := coeff(coeff(f(x, y), y, 1), x, 0);
F := tcoeff(f(x, y));
expand(B^2 - ((4*A) . C));
digits := 20

f := proc (x, y) options operator, arrow; lhs(G) end proc

1, -0.7320831383, 1.326278064, 0.04361098108, 0.4810001561,

-2.656083763

A, B, C, D, E, F := 1, -0.7320831383, 1.326278064, 0.04361098108,

0.4810001561, -2.656083763

A := 0.04361098108

B := 0.4810001561

C := 1.326278064

D := -0.7320831383

E := -2.656083763

F := 1

0.

with(geometry):
_EnvHorizontalName := 'x': _EnvVerticalName := 'y':
conic(co,f(x,y),[x,y]):
detail(co);
/
GeometryDetail\["name of the object", co],
["form of the object", ellipse2d],
["center", [1.212351672 ^(10, 10), -2.198412833 ^(10, 9)]],
["foci", [[2.424703344 ^(10, 10), -4.396825668 ^(10, 9)],
[0.1787052775, 0.9855002601]]],
["length of the major axis", 2.464245740 ^(10, 10)],
["length of the minor axis", 66579.62094],
[                                               2
["equation of the ellipse", 1. + 0.04361098108 x
2
+ 0.4810001561 x y + 1.326278064 y  - 0.7320831383 x
]\
- 2.656083763 y = 0]/;
"_noterminate";

G is a parabola with B^2-4*A*C=0 or an ellipse ? A =1 or F=1 ? Thank you for youy answer.

## Axis and focus of a parabola ...

Maple 2021

How to find the axis and focus of a parabola whose equation we know ? Thank you.

## 2 parabolas that pass through 4 cocyclic...

Maple 2021

How to trace the 2 parabolas that pass through 4 cocyclical points. Thank you

## equation from tangent to conical...

Maple 2021

I try to know an equation of the tangent on the point [0.504244923, 0.3781836925] on the hyperbola7*x^2 - 7*y^2 - 12.0*x + 9.0*y + 2.25 - 2*x*y=0.How can I do that? Thank you.

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