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Matemáticas 3 · Tema 15

Ecuación general de la circunferencia

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Ejemplo

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

Con el centro C=(h,k)C = (h, k) y el radio rr se arma la ecuación y se desarrollan los binomios.

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

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
{C=(−3,4)r=5\begin{cases} C = (-3, 4) \\ r = 5 \end{cases}
Ejercicio 2
{C=(2,3)r=4\begin{cases} C = (2, 3) \\ r = 4 \end{cases}
Ejercicio 3
{C=(5,−1)r=2\begin{cases} C = (5, -1) \\ r = 2 \end{cases}
Ejercicio 4
{C=(−2,−1)r=3\begin{cases} C = (-2, -1) \\ r = 3 \end{cases}

Respuestas

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Ejercicio 1
C=(−3,4)C = (-3, 4)
r=5r = 5
(x−(−3))2+(y−(4))2=(5)2(x+3)2+(y−4)2=25x2+2(x)(3)+(3)2+y2+2(y)(−4)+(−4)2=25x2+6x+9+y2−8y+16=25x2+y2+6x−8y+9+16−25=0x2+y2+6x−8y+25−25=0\begin{aligned} (x - (-3))^2 + (y - (4))^2 &= (5)^2 \\ (x + 3)^2 + (y - 4)^2 &= 25 \\ x^2 + 2(x)(3) + (3)^2 + y^2 + 2(y)(-4) + (-4)^2 &= 25 \\ x^2 + 6x + 9 + y^2 - 8y + 16 &= 25 \\ x^2 + y^2 + 6x - 8y + 9 + 16 - 25 &= 0 \\ x^2 + y^2 + 6x - 8y + 25 - 25 &= 0 \end{aligned}
x2+y2+6x−8y=0x^2 + y^2 + 6x - 8y = 0

Comprobación

(2)2+(4)2+6(2)−8(4)=04+16+12−32=032−32=00=0\begin{aligned} (2)^2 + (4)^2 + 6(2) - 8(4) &= 0 \\ 4 + 16 + 12 - 32 &= 0 \\ 32 - 32 &= 0 \\ 0 &= 0 \end{aligned}
Ejercicio 2
C=(2,3)C = (2, 3)
r=4r = 4
(x−(2))2+(y−(3))2=(4)2(x−2)2+(y−3)2=16x2+2(x)(−2)+(−2)2+y2+2(y)(−3)+(−3)2=16x2−4x+4+y2−6y+9=16x2+y2−4x−6y+4+9−16=0x2+y2−4x−6y+13−16=0\begin{aligned} (x - (2))^2 + (y - (3))^2 &= (4)^2 \\ (x - 2)^2 + (y - 3)^2 &= 16 \\ x^2 + 2(x)(-2) + (-2)^2 + y^2 + 2(y)(-3) + (-3)^2 &= 16 \\ x^2 - 4x + 4 + y^2 - 6y + 9 &= 16 \\ x^2 + y^2 - 4x - 6y + 4 + 9 - 16 &= 0 \\ x^2 + y^2 - 4x - 6y + 13 - 16 &= 0 \end{aligned}
x2+y2−4x−6y−3=0x^2 + y^2 - 4x - 6y - 3 = 0

Comprobación

(2)2+(7)2−4(2)−6(7)−3=04+49−8−42−3=053−53=00=0\begin{aligned} (2)^2 + (7)^2 - 4(2) - 6(7) - 3 &= 0 \\ 4 + 49 - 8 - 42 - 3 &= 0 \\ 53 - 53 &= 0 \\ 0 &= 0 \end{aligned}
Ejercicio 3
C=(5,−1)C = (5, -1)
r=2r = 2
(x−(5))2+(y−(−1))2=(2)2(x−5)2+(y+1)2=4x2+2(x)(−5)+(−5)2+y2+2(y)(1)+(1)2=4x2−10x+25+y2+2y+1=4x2+y2−10x+2y+25+1−4=0x2+y2−10x+2y+26−4=0\begin{aligned} (x - (5))^2 + (y - (-1))^2 &= (2)^2 \\ (x - 5)^2 + (y + 1)^2 &= 4 \\ x^2 + 2(x)(-5) + (-5)^2 + y^2 + 2(y)(1) + (1)^2 &= 4 \\ x^2 - 10x + 25 + y^2 + 2y + 1 &= 4 \\ x^2 + y^2 - 10x + 2y + 25 + 1 - 4 &= 0 \\ x^2 + y^2 - 10x + 2y + 26 - 4 &= 0 \end{aligned}
x2+y2−10x+2y+22=0x^2 + y^2 - 10x + 2y + 22 = 0

Comprobación

(5)2+(−3)2−10(5)+2(−3)+22=025+9−50−6+22=056−56=00=0\begin{aligned} (5)^2 + (-3)^2 - 10(5) + 2(-3) + 22 &= 0 \\ 25 + 9 - 50 - 6 + 22 &= 0 \\ 56 - 56 &= 0 \\ 0 &= 0 \end{aligned}
Ejercicio 4
C=(−2,−1)C = (-2, -1)
r=3r = 3
(x−(−2))2+(y−(−1))2=(3)2(x+2)2+(y+1)2=9x2+2(x)(2)+(2)2+y2+2(y)(1)+(1)2=9x2+4x+4+y2+2y+1=9x2+y2+4x+2y+4+1−9=0x2+y2+4x+2y+5−9=0\begin{aligned} (x - (-2))^2 + (y - (-1))^2 &= (3)^2 \\ (x + 2)^2 + (y + 1)^2 &= 9 \\ x^2 + 2(x)(2) + (2)^2 + y^2 + 2(y)(1) + (1)^2 &= 9 \\ x^2 + 4x + 4 + y^2 + 2y + 1 &= 9 \\ x^2 + y^2 + 4x + 2y + 4 + 1 - 9 &= 0 \\ x^2 + y^2 + 4x + 2y + 5 - 9 &= 0 \end{aligned}
x2+y2+4x+2y−4=0x^2 + y^2 + 4x + 2y - 4 = 0

Comprobación

(−2)2+(2)2+4(−2)+2(2)−4=04+4−8+4−4=012−12=00=0\begin{aligned} (-2)^2 + (2)^2 + 4(-2) + 2(2) - 4 &= 0 \\ 4 + 4 - 8 + 4 - 4 &= 0 \\ 12 - 12 &= 0 \\ 0 &= 0 \end{aligned}

Ejercicios extra

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