Calculo 2 De dos variables_9na Edición - Ron Larson

A x Eje
Pa r aParábola
b o l a
A x
Pa r a bF o lcua
sFoco
d 2 d 2
(x, y) (x, y)
p pd F o cu 1 s
d 1 d 2 d
V e r t e x d 2
Vértice 1 d d (x, 2 y)
p
1
d 1
Di r eDirectriz
c t r i x
d
V e r t e x d 2
1
Figure Figura 10.3 10.3
Di r e c t r i x
Figure 10.3
−2
1
2
1
2
y = − x − x 2
−2
y = V
1ér − t x i − c e
x2
2
)
1
p = − V 1
2 −1,
ér t i c e
2)
1
Foco
1
p = −
1
2 −1, )
−1
2
Foco
−1
1
−1
Parabola with a vertical axis, p < 0
Figure 10.4
Parábola Parabola con with eje a vertical,
axis, p < p 0< 0
Figura Figure 10.4 10.4
1
2
)
y
−1
y
x
x
Parabolas Parábolas
SECCIÓN 10.1 Conics 10.1 and Cónicas Calculus y cálculo 697 697
A Una parabola parábola is the es el set conjunto of all points de todos x, ylos that puntos are equidistant (x, y) equidistantes from a fixed de una line recta called fija llamada
directrix directriz and y a de fixed un punto point fijo, called fuera the de focus dicha not recta, on the llamado line. The foco. midpoint El punto between medio entre
Parabolas
the
the el foco focus A parabola y and directriz the directrix is the es set el is of vértice, the all vertex, points y la and recta x, ythe que that line pasa are passing equidistant por el through foco from y the el vértice a focus fixed and es line el the called eje de
vertex la parábola. the is directrix the Obsérvese axis and of the a fixed en parabola. figura point Note called 10.3 in que the Figure la focus parábola 10.3 not on that es the simétrica a line. parabola The respecto midpoint is symmetric de between su eje.
with the respect focus to and its the axis. directrix is the vertex, and the line passing through the focus and the
vertex is the axis of the parabola. Note in Figure 10.3 that a parabola is symmetric
TEOREMA
with respect
10.1
to
ECUACIÓN
its axis.
ESTÁNDAR O CANÓNICA DE UNA PARÁBOLA
THEOREM 10.1 STANDARD EQUATION OF A PARABOLA
La forma estándar o canónica de la ecuación de una parábola con vértice
The (h, k) standard form of the equation of a parabola with vertex h, k and
THEOREM
y directriz
10.1
y k
STANDARD
p es
EQUATION OF A PARABOLA
directrix y k p is
x
The
standard
h 2 4p
x h 2 form
y k.
Eje vertical.
of the equation of a parabola with vertex h, k and
4py k.
Vertical axis
Para directrix la directriz y x k h p isp,
la ecuación es
For directrix x h p, the equation is
y xk 2 h 2 4px4py h. k.
Eje horizontal. Vertical axis
y k 2 4px h.
Horizontal axis
El foco For se directrix encuentra x en hel eje p, the a p equation unidades is (distancia dirigida) del vértice. Las
The coordenadas focus lies on the axis p units (directed distance) from the vertex. The
y
del
k
foco
Horizontal axis
coordinates of the 2 4px
son las
h.
siguientes.
focus are as follows.
h,
The
k
focus
p
Eje vertical.
lies on the axis p units (directed distance) from the vertex. The
h, k p
Vertical axis
h coordinates p, k of the focus are as follows. Eje horizontal.
h p, k
Horizontal axis
h, k p
Vertical axis
h p, k
Horizontal axis
EXAMPLE EJEMPLO 1 1Hallar Finding el the foco Focus de una of a parábola Parabola
10.1 Conics and Calculus 697
Find Hallar EXAMPLE the el focus foco of de the la 1parábola parabola Finding dada given the por Focus by y of 1 a Parabola 1
x
2 2 x2 .
1 1
Solution Solución Find the To Para focus find hallar the of the focus, el parabola foco, convert se convierte given to standard by a yla forma xby canónica completing o estándar the square. completando el
2 2 x2 .
cuadrado.
1 1
y 2 x
Write original equation.
Solution To find the focus, 2x 2
convert to standard form by completing the square.
1
1 2 x 1 1
y 2 1 2x
2 x2 x 2
Factor Reescribir out
1 1
2. la ecuación original.
y
2y 1
y 2 x
2x x
21 2x 2 2x 2
Write original equation.
x 2 Multiply each side by 2.
1
Sacar como factor. 1
y
2y 1
2 1
x 2 2x x
2x
2
Factor out
Group terms.
2.
2y 2y 1 12x x Multiplicar cada lado por 2.
2y 2 x 2 2x 2 x
2x 2 Multiply each side by 2.
1 Add and subtract 1 on right side.
x 2 2y 2y 1 1x x 2x 1 2y 2
2 2x 2x Agrupar Group términos. terms.
x 12y 2 2y 2
2 x 2 x
2 y 1 2 2x 1 Add and subtract 1 on right side.
2x 1 Write Sumar in y standard restar 1 form. en el lado derecho.
x 2 2x 1 2y 2
Comparing x 2 2x this equation 1 2ywith
2x h 2 4py k, you can conclude that
x 1 2 2 y 1
Write in standard form.
x 1 2 1
h 1, k2y 1, and
1 p Expresar en la forma estándar o canónica.
Comparing this equation with x h 2 2.
4py k, you can conclude that
Because p is negative, the parabola opens
Si se compara esta ecuación con x h 2 downward, 1 as shown in Figure 10.4. So, the
h 1, k 1, and p 4p y k, se concluye que
focus of the parabola is p units from the vertex, or2.
Because h 1, p is negative, k y p 1 2
h, k p 1, 1 the parabola opens . downward, as shown in Figure 10.4. So, the
2 .
Focus
focus of the parabola is p units from the vertex, or
Como p es negativo, la parábola se abre hacia abajo, como se muestra en la figura 10.4.
Por tanto, h, el kfoco pde la parábola 1, 1 se encuentra a p unidades del vértice, o sea
A line segment that passes
2 .
Focus
through the focus of a parabola and has endpoints on
the parabola h, k is pcalled 1, a focal 1 2. chord. The specific Foco. focal chord perpendicular to the axis
of the parabola A line segment is the latus that rectum. passes through The next the example focus of shows a parabola how to and determine has endpoints the on
length the of parabola the latus is rectum called a and focal the chord. length The of the specific corresponding focal chord intercepted perpendicular arc. to the axis
of the parabola is the latus rectum. The next example shows how to determine the
A un segmento de la recta que pasa por el foco de una parábola y que tiene sus extremos
en la parábola se le llama cuerda focal. La cuerda focal perpendicular al eje de la
length of the latus rectum and the length of the corresponding intercepted arc.
parábola es el lado recto (latus rectum). El ejemplo siguiente muestra cómo determinar
la longitud del lado recto y la longitud del correspondiente arco cortado.