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Matemáticas 3 · Tema 17

Ecuación general de la parábola

Ejemplo

(x−h)2=±4p(y−k)(x - h)^2 = \pm 4p(y - k)

Con el signo + la parábola abre hacia arriba; con el signo −, hacia abajo. pp es la distancia del vértice al foco.

foco(−4,3)(-4, 3)
directrizy=−1y = -1
vértice(−4,1)(-4, 1)
p22
(x−(−4))2=+4(2)(y−(1))(x+4)2=+8(y−1)x2+8x+16=8y−8x2+8x+16−8y+8=0\begin{aligned} (x - (-4))^2 &= +4(2)(y - (1)) \\ (x + 4)^2 &= +8(y - 1) \\ x^2 + 8x + 16 &= 8y - 8 \\ x^2 + 8x + 16 - 8y + 8 &= 0 \end{aligned}
x2+8x−8y+24=0x^2 + 8x - 8y + 24 = 0

Comprobación

(−8)2+8(−8)−8(3)+24=064−64−24+24=00=0\begin{aligned} (-8)^2 + 8(-8) - 8(3) + 24 &= 0 \\ 64 - 64 - 24 + 24 &= 0 \\ 0 &= 0 \end{aligned}
(0)2+8(0)−8(3)+24=00+0−24+24=00=0\begin{aligned} (0)^2 + 8(0) - 8(3) + 24 &= 0 \\ 0 + 0 - 24 + 24 &= 0 \\ 0 &= 0 \end{aligned}

Ejercicios

Ejercicio 1
{F=(−4,3)y=−5\begin{cases} F = (-4, 3) \\ y = -5 \end{cases}
Ejercicio 2
{F=(−5,2)y=6\begin{cases} F = (-5, 2) \\ y = 6 \end{cases}
Ejercicio 3
{F=(3,2)y=−4\begin{cases} F = (3, 2) \\ y = -4 \end{cases}
Ejercicio 4
{F=(2,1)y=7\begin{cases} F = (2, 1) \\ y = 7 \end{cases}

Respuestas

▼
Ejercicio 1
foco(−4,3)(-4, 3)
directrizy=−5y = -5
vértice(−4,−1)(-4, -1)
p44
(x−(−4))2=+4(4)(y−(−1))(x+4)2=+16(y+1)x2+8x+16=16y+16x2+8x+16−16y−16=0\begin{aligned} (x - (-4))^2 &= +4(4)(y - (-1)) \\ (x + 4)^2 &= +16(y + 1) \\ x^2 + 8x + 16 &= 16y + 16 \\ x^2 + 8x + 16 - 16y - 16 &= 0 \end{aligned}
x2+8x−16y=0x^2 + 8x - 16y = 0

Comprobación

(−12)2+8(−12)−16(3)=0144−96−48=0144−144=00=0\begin{aligned} (-12)^2 + 8(-12) - 16(3) &= 0 \\ 144 - 96 - 48 &= 0 \\ 144 - 144 &= 0 \\ 0 &= 0 \end{aligned}
(4)2+8(4)−16(3)=016+32−48=048−48=00=0\begin{aligned} (4)^2 + 8(4) - 16(3) &= 0 \\ 16 + 32 - 48 &= 0 \\ 48 - 48 &= 0 \\ 0 &= 0 \end{aligned}
Ejercicio 2
foco(−5,2)(-5, 2)
directrizy=6y = 6
vértice(−5,4)(-5, 4)
p22
(x−(−5))2=−4(2)(y−(4))(x+5)2=−8(y−4)x2+10x+25=−8y+32x2+10x+25+8y−32=0\begin{aligned} (x - (-5))^2 &= -4(2)(y - (4)) \\ (x + 5)^2 &= -8(y - 4) \\ x^2 + 10x + 25 &= -8y + 32 \\ x^2 + 10x + 25 + 8y - 32 &= 0 \end{aligned}
x2+10x+8y−7=0x^2 + 10x + 8y - 7 = 0

Comprobación

(−9)2+10(−9)+8(2)−7=081−90+16−7=097−97=00=0\begin{aligned} (-9)^2 + 10(-9) + 8(2) - 7 &= 0 \\ 81 - 90 + 16 - 7 &= 0 \\ 97 - 97 &= 0 \\ 0 &= 0 \end{aligned}
(−1)2+10(−1)+8(2)−7=01−10+16−7=017−17=00=0\begin{aligned} (-1)^2 + 10(-1) + 8(2) - 7 &= 0 \\ 1 - 10 + 16 - 7 &= 0 \\ 17 - 17 &= 0 \\ 0 &= 0 \end{aligned}
Ejercicio 3
foco(3,2)(3, 2)
directrizy=−4y = -4
vértice(3,−1)(3, -1)
p33
(x−(3))2=+4(3)(y−(−1))(x−3)2=12(y+1)x2−6x+9=12y+12x2−6x+9−12y−12=0\begin{aligned} (x - (3))^2 &= +4(3)(y - (-1)) \\ (x - 3)^2 &= 12(y + 1) \\ x^2 - 6x + 9 &= 12y + 12 \\ x^2 - 6x + 9 - 12y - 12 &= 0 \end{aligned}
x2−6x−12y−3=0x^2 - 6x - 12y - 3 = 0

Comprobación

(−3)2−6(−3)−12(2)−3=09+18−24−3=027−27=00=0\begin{aligned} (-3)^2 - 6(-3) - 12(2) - 3 &= 0 \\ 9 + 18 - 24 - 3 &= 0 \\ 27 - 27 &= 0 \\ 0 &= 0 \end{aligned}
(9)2−6(9)−12(2)−3=081−54−24−3=081−81=00=0\begin{aligned} (9)^2 - 6(9) - 12(2) - 3 &= 0 \\ 81 - 54 - 24 - 3 &= 0 \\ 81 - 81 &= 0 \\ 0 &= 0 \end{aligned}
Ejercicio 4
foco(2,1)(2, 1)
directrizy=7y = 7
vértice(2,4)(2, 4)
p33
(x−(2))2=−4(3)(y−(4))(x−2)2=−12(y−4)x2−4x+4=−12y+48x2−4x+4+12y−48=0\begin{aligned} (x - (2))^2 &= -4(3)(y - (4)) \\ (x - 2)^2 &= -12(y - 4) \\ x^2 - 4x + 4 &= -12y + 48 \\ x^2 - 4x + 4 + 12y - 48 &= 0 \end{aligned}
x2−4x+12y−44=0x^2 - 4x + 12y - 44 = 0

Comprobación

(−4)2−4(−4)+12(1)−44=016+16+12−44=044−44=00=0\begin{aligned} (-4)^2 - 4(-4) + 12(1) - 44 &= 0 \\ 16 + 16 + 12 - 44 &= 0 \\ 44 - 44 &= 0 \\ 0 &= 0 \end{aligned}
(8)2−4(8)+12(1)−44=064−32+12−44=076−76=00=0\begin{aligned} (8)^2 - 4(8) + 12(1) - 44 &= 0 \\ 64 - 32 + 12 - 44 &= 0 \\ 76 - 76 &= 0 \\ 0 &= 0 \end{aligned}

Ejercicios extra

▼
Bloque 1
{F=(2,5)y=1\begin{cases} F = (2, 5) \\ y = 1 \end{cases}
{F=(−3,1)y=5\begin{cases} F = (-3, 1) \\ y = 5 \end{cases}
{F=(0,4)y=−2\begin{cases} F = (0, 4) \\ y = -2 \end{cases}
{F=(4,−1)y=3\begin{cases} F = (4, -1) \\ y = 3 \end{cases}
{F=(−2,6)y=2\begin{cases} F = (-2, 6) \\ y = 2 \end{cases}
Bloque 2
{F=(5,0)y=4\begin{cases} F = (5, 0) \\ y = 4 \end{cases}
{F=(−1,−2)y=−6\begin{cases} F = (-1, -2) \\ y = -6 \end{cases}
{F=(3,7)y=1\begin{cases} F = (3, 7) \\ y = 1 \end{cases}
{F=(−5,3)y=7\begin{cases} F = (-5, 3) \\ y = 7 \end{cases}
{F=(1,−4)y=2\begin{cases} F = (1, -4) \\ y = 2 \end{cases}
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