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Matemáticas
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Semestre entero de la asignatura matemáticas I
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MATEMÁTICASI
,IEIYAI :(ÓNICASY (UÁDR
-
÷.mm
Las
CÓNICAS importan
cónicas más
elipses e hipérbolas son
DEGENERADAS . El resto so
Se define como :
'
axl +
by +
cxy + dx + f. = O
ey +
PARÁBOLA
[aso1:q=k DCP F) =D CP ,
,
recta directriz )
Fiyi
µ
=
y +
L
Elevando al cuadrado :
y.yyyg.gg ya g. , + ↳y →
Foco
( o /
÷)
Caso 2 : ✗ =
ky
'
K> o abierta ha
OX vértice ( 0,0 )
kco abierta ha
,ELIPSE
Ecuación reducida :
÷ 4¥ + = 1 a
* Como A está en la elipse :
d (A) F) 1- d ( A
,
/ a- f) + la f) + = K
* Como B está en la eli
Ld (B. F) = 2a → d CB ,
"
'
'
b
f = ar -
Pcx está la
,
y)
en
elipse , por tanto
:
lx-ifj-yf-ifx-fj.ly#=2al(xi-fT-TT=(2a-Fg-yY
desarrollando y simplificando :
✗
f- al = - a
Ég ÷ ¥- +
* Si a = b → se trata de una circunferencia de centro (op )
a
, signo contrario
&÷
2
✗
* si a > b →
-
+ = 1 f- = á -
b
ai
centro ( 0,0)
vértices ( a , o) , f- a , o) ,
( o , b) , 10 ,
-
b)
f
a > b :
focos ( f ,
o ) ,
( fi
-
e =
ób má
b) a :
focos ( o , f) , (o ,
f)
-
a
HIPÉRBOLAS
CASOF Ecuación reducida :
÷ ÷ -
distancia entre focos f
l =
a
-
=
distancia entre vértices
ya
€2
-
Ecuación reducida :
J
-
distancia entre focos
e =
distancia entre vértices
=
÷
ECUACIÓN REDUCIDA DE UNA CÓNICA
• Si C =/ O → girada
•
Si c. = o → no girada
CÓNICAS TRASLADADAS → C = O
'
AXZ +
by + dx f--0
+
ey +
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