Inicio › Matemáticas 3 › Regreso de la circunferencia

Matemáticas 3 · Tema 16

Regreso de la circunferencia

Video

Ejemplo

A partir de la ecuación general se encuentran el centro y el radio completando los trinomios cuadrados perfectos.

x2+y2+2x−4y−20=0x^2 + y^2 + 2x - 4y - 20 = 0
b22
mitad11
mitad²11
b−4-4
mitad−2-2
mitad²44
x2+2x+y2−4y=20x2+2x+1+y2−4y+4=20+1+4(x+1)2+(y−2)2=25(x+1)2+(y−2)2=(5)2(x−(−1))2+(y−(2))2=(5)2\begin{aligned} x^2 + 2x + y^2 - 4y &= 20 \\ x^2 + 2x + 1 + y^2 - 4y + 4 &= 20 + 1 + 4 \\ (x + 1)^2 + (y - 2)^2 &= 25 \\ (x + 1)^2 + (y - 2)^2 &= (5)^2 \\ (x - (-1))^2 + (y - (2))^2 &= (5)^2 \end{aligned}
C=(−1,2)C = (-1, 2)
r=5r = 5

Comprobación

(−6)2+(2)2+2(−6)−4(2)−20=036+4−12−8−20=040−40=00=0\begin{aligned} (-6)^2 + (2)^2 + 2(-6) - 4(2) - 20 &= 0 \\ 36 + 4 - 12 - 8 - 20 &= 0 \\ 40 - 40 &= 0 \\ 0 &= 0 \end{aligned}

Ejercicios

Ejercicio 1
x2+y2−8x+6y+16=0x^2 + y^2 - 8x + 6y + 16 = 0
Ejercicio 2
x2+y2−6x+10y+33=0x^2 + y^2 - 6x + 10y + 33 = 0
Ejercicio 3
x2+y2+2x−10y+17=0x^2 + y^2 + 2x - 10y + 17 = 0
Ejercicio 4
x2+y2−4x−6y−12=0x^2 + y^2 - 4x - 6y - 12 = 0

Respuestas

▼
Ejercicio 1
x2+y2−8x+6y+16=0x^2 + y^2 - 8x + 6y + 16 = 0
b−8-8
mitad−4-4
mitad²1616
b66
mitad33
mitad²99
x2−8x+y2+6y=−16x2−8x+16+y2+6y+9=−16+16+9(x−4)2+(y+3)2=9(x−4)2+(y+3)2=(3)2(x−(4))2+(y−(−3))2=(3)2\begin{aligned} x^2 - 8x + y^2 + 6y &= -16 \\ x^2 - 8x + 16 + y^2 + 6y + 9 &= -16 + 16 + 9 \\ (x - 4)^2 + (y + 3)^2 &= 9 \\ (x - 4)^2 + (y + 3)^2 &= (3)^2 \\ (x - (4))^2 + (y - (-3))^2 &= (3)^2 \end{aligned}
C=(4,−3)C = (4, -3)
r=3r = 3

Comprobación

(4)2+(0)2−8(4)+6(0)+16=016+0−32+0+16=032−32=00=0\begin{aligned} (4)^2 + (0)^2 - 8(4) + 6(0) + 16 &= 0 \\ 16 + 0 - 32 + 0 + 16 &= 0 \\ 32 - 32 &= 0 \\ 0 &= 0 \end{aligned}
Ejercicio 2
x2+y2−6x+10y+33=0x^2 + y^2 - 6x + 10y + 33 = 0
b−6-6
mitad−3-3
mitad²99
b1010
mitad55
mitad²2525
x2−6x+y2+10y=−33x2−6x+9+y2+10y+25=−33+9+25(x−3)2+(y+5)2=1(x−3)2+(y+5)2=(1)2(x−(3))2+(y−(−5))2=(1)2\begin{aligned} x^2 - 6x + y^2 + 10y &= -33 \\ x^2 - 6x + 9 + y^2 + 10y + 25 &= -33 + 9 + 25 \\ (x - 3)^2 + (y + 5)^2 &= 1 \\ (x - 3)^2 + (y + 5)^2 &= (1)^2 \\ (x - (3))^2 + (y - (-5))^2 &= (1)^2 \end{aligned}
C=(3,−5)C = (3, -5)
r=1r = 1

Comprobación

(4)2+(−5)2−6(4)+10(−5)+33=016+25−24−50+33=074−74=00=0\begin{aligned} (4)^2 + (-5)^2 - 6(4) + 10(-5) + 33 &= 0 \\ 16 + 25 - 24 - 50 + 33 &= 0 \\ 74 - 74 &= 0 \\ 0 &= 0 \end{aligned}
Ejercicio 3
x2+y2+2x−10y+17=0x^2 + y^2 + 2x - 10y + 17 = 0
b22
mitad11
mitad²11
b−10-10
mitad−5-5
mitad²2525
x2+2x+y2−10y=−17x2+2x+1+y2−10y+25=−17+1+25(x+1)2+(y−5)2=9(x+1)2+(y−5)2=(3)2(x−(−1))2+(y−(5))2=(3)2\begin{aligned} x^2 + 2x + y^2 - 10y &= -17 \\ x^2 + 2x + 1 + y^2 - 10y + 25 &= -17 + 1 + 25 \\ (x + 1)^2 + (y - 5)^2 &= 9 \\ (x + 1)^2 + (y - 5)^2 &= (3)^2 \\ (x - (-1))^2 + (y - (5))^2 &= (3)^2 \end{aligned}
C=(−1,5)C = (-1, 5)
r=3r = 3

Comprobación

(−1)2+(2)2+2(−1)−10(2)+17=01+4−2−20+17=022−22=00=0\begin{aligned} (-1)^2 + (2)^2 + 2(-1) - 10(2) + 17 &= 0 \\ 1 + 4 - 2 - 20 + 17 &= 0 \\ 22 - 22 &= 0 \\ 0 &= 0 \end{aligned}
Ejercicio 4
x2+y2−4x−6y−12=0x^2 + y^2 - 4x - 6y - 12 = 0
b−4-4
mitad−2-2
mitad²44
b−6-6
mitad−3-3
mitad²99
x2−4x+y2−6y=12x2−4x+4+y2−6y+9=12+4+9(x−2)2+(y−3)2=25(x−2)2+(y−3)2=(5)2(x−(2))2+(y−(3))2=(5)2\begin{aligned} x^2 - 4x + y^2 - 6y &= 12 \\ x^2 - 4x + 4 + y^2 - 6y + 9 &= 12 + 4 + 9 \\ (x - 2)^2 + (y - 3)^2 &= 25 \\ (x - 2)^2 + (y - 3)^2 &= (5)^2 \\ (x - (2))^2 + (y - (3))^2 &= (5)^2 \end{aligned}
C=(2,3)C = (2, 3)
r=5r = 5

Comprobación

(2)2+(8)2−4(2)−6(8)−12=04+64−8−48−12=068−68=00=0\begin{aligned} (2)^2 + (8)^2 - 4(2) - 6(8) - 12 &= 0 \\ 4 + 64 - 8 - 48 - 12 &= 0 \\ 68 - 68 &= 0 \\ 0 &= 0 \end{aligned}

Ejercicios extra

▼
Bloque 1
x2+y2−2x−8y+8=0x^2 + y^2 - 2x - 8y + 8 = 0
x2+y2+6x−4y−3=0x^2 + y^2 + 6x - 4y - 3 = 0
x2+y2−10x+4y−7=0x^2 + y^2 - 10x + 4y - 7 = 0
x2+y2+8x+2y+13=0x^2 + y^2 + 8x + 2y + 13 = 0
x2+y2−6y−16=0x^2 + y^2 - 6y - 16 = 0
Bloque 2
x2+y2−4x−12y+24=0x^2 + y^2 - 4x - 12y + 24 = 0
x2+y2+2x+10y−23=0x^2 + y^2 + 2x + 10y - 23 = 0
x2+y2−12x−2y+12=0x^2 + y^2 - 12x - 2y + 12 = 0
x2+y2+4x−8y−16=0x^2 + y^2 + 4x - 8y - 16 = 0
x2+y2−6x+8y+21=0x^2 + y^2 - 6x + 8y + 21 = 0
← Ecuación de la circunferencia Ecuación de la parábola →