formulaire coniques 1
L’ellipse
Ellipse centrée en O(0, 0)
Axe focal Ox
Axe focal Oy
y2
x2
=1
a2
b2
y2
x2
=1
b2
a2
a>b
a<b
Sommets
A(a, 0)
A0 (−a, 0)
B(0, b)
B 0 (0, −b)
A(0, a)
A0 (0, −a)
B(b, 0)
B 0 (−b, 0)
Foyers
F (c, 0)
F 0 (−c, 0)
F (0, c)
F 0 (0, −c)
c2 = a2 − b2
Tangente au point A(xA , yA )
xxA
yyA
2 =1
a2
b
yyA
xxA
2 =1
a2
b
Ellipse centrée en C(xC , yC )
Axe focal Ox
Axe focal Oy
(x − xC )2
(y − yC )2
=1
2
a
b2
(x − xC )2
(yy c)2
=1
2
b
a2
a>b
a<b
Sommets
A(a xC , yC )
A0 (−a xC , yC )
B(xC , b yC )
B 0 (xC , −b yC )
A(xC , a yC )
A0 (xC , −a yC )
B(b xC , yC )
B 0 (−bxC , yC )
Foyers
F (c xC , yC )
F 0 (−c xC , yC )
F (xC , c yC )
F 0 (xC , −c yC )
c2 = a2 − b2
Tangente au point A(xA , yA )
(x − xC )xA
(y − yC )yA
=1
2
a
b2
(y − yC )yA
(x − xC )xA
=1
2
a
b2
1
2
L’hyperbole
Hyperbole centrée en O(0, 0)
Axe focal Ox
Axe focal Oy
y2
x2
−
=1
a2
b2
x2
y2
−
=1
a2
b2
Sommets
A(a, 0)
A0 (−a, 0)
A(0, a)
A0 (0, −a)
Foyers
F (c, 0)
F 0 (−c, 0)
F (0, c)
F 0 (0, −c)
c2 = a2 b2
Asymptotes
b
AO ≡ y = ± x
a
Tangente au point A(xA , yA )
xxA
yyA
− 2 =1
2
a
b
a
AO ≡ y = ± x
b
yyA
xxA
− 2 =1
2
a
b
Hyperbole centrée en C(xC , yC )
Axe focal Ox
Axe focal Oy
(x − xC )2
(y − yC )2
−
=1
a2
b2
(y − yC )2
(x − xC )2
−
=1
a2
b2
Sommets
A(a xC , yC )
A0 (−a xC , yC )
A(xC , a yC )
A0 (xC , −a yC )
Foyers
F (c xC , yC )
F 0 (−c xC , yC )
F (xC , c yC )
F 0 (xC , −c yC )
c2 = a2 − b2
Asymptotes
b
a
AO ≡ (y − yC ) = ± (x − xC )
AO ≡ (y − yC ) = ± (x − xC )
a
b
Tangente au point A(xA , yA )
(x − xC )xA
(y − yC )yA
−
=1
a2
b2
(y − yC )yA
(x − xC )xA
−
=1
a2
b2
2
3
La parabole
Parabole de sommet O(0, 0)
Axe focal Ox
Axe focal Oy
y 2 = 2px
x2 = 2py
p >