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La ecuación:
Ecuación 5*x^2+2*x-7=0
Ecuación 12,4-x=16
Ecuación x+120=40*5
Ecuación y(0)=1
Expresar {x} en función de y en la ecuación:
-18*x-8*y=-18
-13*x+12*y=-19
-1*x+11*y=-9
16*x-11*y=13
Integral de d{x}:
r^2
Expresiones idénticas
(x-x0)^ dos +(y-y0)^ dos =r^ dos
(x menos x0) al cuadrado más (y menos y0) al cuadrado es igual a r al cuadrado
(x menos x0) en el grado dos más (y menos y0) en el grado dos es igual a r en el grado dos
(x-x0)2+(y-y0)2=r2
x-x02+y-y02=r2
(x-x0)²+(y-y0)²=r²
(x-x0) en el grado 2+(y-y0) en el grado 2=r en el grado 2
x-x0^2+y-y0^2=r^2
Expresiones semejantes
(x+x0)^2+(y-y0)^2=r^2
(x-x0)^2-(y-y0)^2=r^2
(x-x0)^2+(y+y0)^2=r^2

(x-x0)^2+(y-y0)^2=r^2 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

Solución

Ha introducido [src]

        2           2    2
(x - x0)  + (y - y0)  = r

$$\left(x - x_{0}\right)^{2} + \left(y - y_{0}\right)^{2} = r^{2}$$

Simplificar

Solución detallada

Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.

La ecuación se convierte de
$$\left(x - x_{0}\right)^{2} + \left(y - y_{0}\right)^{2} = r^{2}$$
en
$$- r^{2} + \left(\left(x - x_{0}\right)^{2} + \left(y - y_{0}\right)^{2}\right) = 0$$
Abramos la expresión en la ecuación
$$- r^{2} + \left(\left(x - x_{0}\right)^{2} + \left(y - y_{0}\right)^{2}\right) = 0$$
Obtenemos la ecuación cuadrática
$$- r^{2} + x^{2} - 2 x x_{0} + x_{0}^{2} + y^{2} - 2 y y_{0} + y_{0}^{2} = 0$$
Es la ecuación de la forma

a*x^2 + b*x + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = - 2 x_{0}$$
$$c = - r^{2} + x_{0}^{2} + y^{2} - 2 y y_{0} + y_{0}^{2}$$
, entonces

D = b^2 - 4 * a * c =

(-2*x0)^2 - 4 * (1) * (x0^2 + y^2 + y0^2 - r^2 - 2*y*y0) = -4*y^2 - 4*y0^2 + 4*r^2 + 8*y*y0

La ecuación tiene dos raíces.

x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

o
$$x_{1} = x_{0} + \frac{\sqrt{4 r^{2} - 4 y^{2} + 8 y y_{0} - 4 y_{0}^{2}}}{2}$$
$$x_{2} = x_{0} - \frac{\sqrt{4 r^{2} - 4 y^{2} + 8 y y_{0} - 4 y_{0}^{2}}}{2}$$

Gráfica

Respuesta rápida [src]

       /      _________________________________________________________________________________________________________________________________________                                                                                                                                                      \       _________________________________________________________________________________________________________________________________________                                                                                                                                                      
       |     /                                                                                                                                       2     /     /                                                                2        2         2        2        2        2                 \\         |      /                                                                                                                                       2     /     /                                                                2        2         2        2        2        2                 \\         
       |  4 /                                                                2   /  2        2         2        2        2        2                 \      |atan2\2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r), im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/|         |   4 /                                                                2   /  2        2         2        2        2        2                 \      |atan2\2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r), im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/|         
x1 = I*|- \/   (2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r))  + \im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/  *sin|---------------------------------------------------------------------------------------------------------------------------------------| + im(x0)| - \/   (2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r))  + \im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/  *cos|---------------------------------------------------------------------------------------------------------------------------------------| + re(x0)
       \                                                                                                                                                   \                                                                   2                                                                   /         /                                                                                                                                                    \                                                                   2                                                                   /

$$x_{1} = i \left(- \sqrt[4]{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)},\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(x_{0}\right)}\right) - \sqrt[4]{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)},\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(x_{0}\right)}$$

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       |   /                                                                                                                                       2     /     /                                                                2        2         2        2        2        2                 \\         |      /                                                                                                                                       2     /     /                                                                2        2         2        2        2        2                 \\         
       |4 /                                                                2   /  2        2         2        2        2        2                 \      |atan2\2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r), im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/|         |   4 /                                                                2   /  2        2         2        2        2        2                 \      |atan2\2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r), im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/|         
x2 = I*|\/   (2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r))  + \im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/  *sin|---------------------------------------------------------------------------------------------------------------------------------------| + im(x0)| + \/   (2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r))  + \im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/  *cos|---------------------------------------------------------------------------------------------------------------------------------------| + re(x0)
       \                                                                                                                                                 \                                                                   2                                                                   /         /                                                                                                                                                    \                                                                   2                                                                   /

$$x_{2} = i \left(\sqrt[4]{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)},\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(x_{0}\right)}\right) + \sqrt[4]{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)},\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(x_{0}\right)}$$

x2 = i*(((2*re(r)*im(r) - 2*re(y)*im(y) - 2*re(y0)*im(y0) + 2*im(y*y0))^2 + (re(r)^2 - re(y)^2 - re(y0)^2 + 2*re(y*y0) - im(r)^2 + im(y)^2 + im(y0)^2)^2)^(1/4)*sin(atan2(2*re(r)*im(r) - 2*re(y)*im(y) - 2*re(y0)*im(y0) + 2*im(y*y0, re(r)^2 - re(y)^2 - re(y0)^2 + 2*re(y*y0) - im(r)^2 + im(y)^2 + im(y0)^2)/2) + im(x0)) + ((2*re(r)*im(r) - 2*re(y)*im(y) - 2*re(y0)*im(y0) + 2*im(y*y0))^2 + (re(r)^2 - re(y)^2 - re(y0)^2 + 2*re(y*y0) - im(r)^2 + im(y)^2 + im(y0)^2)^2)^(1/4)*cos(atan2(2*re(r)*im(r) - 2*re(y)*im(y) - 2*re(y0)*im(y0) + 2*im(y*y0), re(r)^2 - re(y)^2 - re(y0)^2 + 2*re(y*y0) - im(r)^2 + im(y)^2 + im(y0)^2)/2) + re(x0))

Suma y producto de raíces [src]

suma

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  |     /                                                                                                                                       2     /     /                                                                2        2         2        2        2        2                 \\         |      /                                                                                                                                       2     /     /                                                                2        2         2        2        2        2                 \\              |   /                                                                                                                                       2     /     /                                                                2        2         2        2        2        2                 \\         |      /                                                                                                                                       2     /     /                                                                2        2         2        2        2        2                 \\         
  |  4 /                                                                2   /  2        2         2        2        2        2                 \      |atan2\2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r), im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/|         |   4 /                                                                2   /  2        2         2        2        2        2                 \      |atan2\2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r), im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/|              |4 /                                                                2   /  2        2         2        2        2        2                 \      |atan2\2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r), im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/|         |   4 /                                                                2   /  2        2         2        2        2        2                 \      |atan2\2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r), im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/|         
I*|- \/   (2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r))  + \im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/  *sin|---------------------------------------------------------------------------------------------------------------------------------------| + im(x0)| - \/   (2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r))  + \im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/  *cos|---------------------------------------------------------------------------------------------------------------------------------------| + re(x0) + I*|\/   (2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r))  + \im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/  *sin|---------------------------------------------------------------------------------------------------------------------------------------| + im(x0)| + \/   (2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r))  + \im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/  *cos|---------------------------------------------------------------------------------------------------------------------------------------| + re(x0)
  \                                                                                                                                                   \                                                                   2                                                                   /         /                                                                                                                                                    \                                                                   2                                                                   /              \                                                                                                                                                 \                                                                   2                                                                   /         /                                                                                                                                                    \                                                                   2                                                                   /

$$\left(i \left(- \sqrt[4]{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)},\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(x_{0}\right)}\right) - \sqrt[4]{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)},\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(x_{0}\right)}\right) + \left(i \left(\sqrt[4]{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)},\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(x_{0}\right)}\right) + \sqrt[4]{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)},\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(x_{0}\right)}\right)$$

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             |   /                                                                                                                                       2     /     /                                                                2        2         2        2        2        2                 \\         |     |     /                                                                                                                                       2     /     /                                                                2        2         2        2        2        2                 \\         |
             |4 /                                                                2   /  2        2         2        2        2        2                 \      |atan2\2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r), im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/|         |     |  4 /                                                                2   /  2        2         2        2        2        2                 \      |atan2\2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r), im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/|         |
2*re(x0) + I*|\/   (2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r))  + \im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/  *sin|---------------------------------------------------------------------------------------------------------------------------------------| + im(x0)| + I*|- \/   (2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r))  + \im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/  *sin|---------------------------------------------------------------------------------------------------------------------------------------| + im(x0)|
             \                                                                                                                                                 \                                                                   2                                                                   /         /     \                                                                                                                                                   \                                                                   2                                                                   /         /

$$i \left(- \sqrt[4]{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)},\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(x_{0}\right)}\right) + i \left(\sqrt[4]{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)},\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(x_{0}\right)}\right) + 2 \operatorname{re}{\left(x_{0}\right)}$$

producto

/  /      _________________________________________________________________________________________________________________________________________                                                                                                                                                      \       _________________________________________________________________________________________________________________________________________                                                                                                                                                      \ /  /    _________________________________________________________________________________________________________________________________________                                                                                                                                                      \       _________________________________________________________________________________________________________________________________________                                                                                                                                                      \
|  |     /                                                                                                                                       2     /     /                                                                2        2         2        2        2        2                 \\         |      /                                                                                                                                       2     /     /                                                                2        2         2        2        2        2                 \\         | |  |   /                                                                                                                                       2     /     /                                                                2        2         2        2        2        2                 \\         |      /                                                                                                                                       2     /     /                                                                2        2         2        2        2        2                 \\         |
|  |  4 /                                                                2   /  2        2         2        2        2        2                 \      |atan2\2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r), im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/|         |   4 /                                                                2   /  2        2         2        2        2        2                 \      |atan2\2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r), im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/|         | |  |4 /                                                                2   /  2        2         2        2        2        2                 \      |atan2\2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r), im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/|         |   4 /                                                                2   /  2        2         2        2        2        2                 \      |atan2\2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r), im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/|         |
|I*|- \/   (2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r))  + \im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/  *sin|---------------------------------------------------------------------------------------------------------------------------------------| + im(x0)| - \/   (2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r))  + \im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/  *cos|---------------------------------------------------------------------------------------------------------------------------------------| + re(x0)|*|I*|\/   (2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r))  + \im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/  *sin|---------------------------------------------------------------------------------------------------------------------------------------| + im(x0)| + \/   (2*im(y*y0) - 2*im(y)*re(y) - 2*im(y0)*re(y0) + 2*im(r)*re(r))  + \im (y) + im (y0) + re (r) - im (r) - re (y) - re (y0) + 2*re(y*y0)/  *cos|---------------------------------------------------------------------------------------------------------------------------------------| + re(x0)|
\  \                                                                                                                                                   \                                                                   2                                                                   /         /                                                                                                                                                    \                                                                   2                                                                   /         / \  \                                                                                                                                                 \                                                                   2                                                                   /         /                                                                                                                                                    \                                                                   2                                                                   /         /

$$\left(i \left(- \sqrt[4]{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)},\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(x_{0}\right)}\right) - \sqrt[4]{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)},\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(x_{0}\right)}\right) \left(i \left(\sqrt[4]{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)},\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(x_{0}\right)}\right) + \sqrt[4]{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} - 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} + 2 \operatorname{im}{\left(y y_{0}\right)},\left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 \operatorname{re}{\left(y y_{0}\right)} - \left(\operatorname{im}{\left(r\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(x_{0}\right)}\right)$$

  2        2         2        2         2         2        2         2                                                                                                           
im (r) + re (x0) + re (y) + re (y0) - im (x0) - im (y) - im (y0) - re (r) - 2*re(y*y0) - 2*I*im(y*y0) - 2*I*im(r)*re(r) + 2*I*im(x0)*re(x0) + 2*I*im(y)*re(y) + 2*I*im(y0)*re(y0)

$$- \left(\operatorname{re}{\left(r\right)}\right)^{2} - 2 i \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)} + \left(\operatorname{re}{\left(x_{0}\right)}\right)^{2} + 2 i \operatorname{re}{\left(x_{0}\right)} \operatorname{im}{\left(x_{0}\right)} + \left(\operatorname{re}{\left(y\right)}\right)^{2} + 2 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + \left(\operatorname{re}{\left(y_{0}\right)}\right)^{2} + 2 i \operatorname{re}{\left(y_{0}\right)} \operatorname{im}{\left(y_{0}\right)} - 2 \operatorname{re}{\left(y y_{0}\right)} + \left(\operatorname{im}{\left(r\right)}\right)^{2} - \left(\operatorname{im}{\left(x_{0}\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y_{0}\right)}\right)^{2} - 2 i \operatorname{im}{\left(y y_{0}\right)}$$

im(r)^2 + re(x0)^2 + re(y)^2 + re(y0)^2 - im(x0)^2 - im(y)^2 - im(y0)^2 - re(r)^2 - 2*re(y*y0) - 2*i*im(y*y0) - 2*i*im(r)*re(r) + 2*i*im(x0)*re(x0) + 2*i*im(y)*re(y) + 2*i*im(y0)*re(y0)

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(x-x0)^2+(y-y0)^2=r^2 la ecuación

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