nyuuzyou/wikis huggingface dataset:
Equation of Circle in Complex Plane/Formulation 2/Proof 2

== Theorem ==
== Proof ==
From Equation of Circle in Cartesian Plane: Formulation 2, the equation for a circle is:
:$A \paren {x^2 + y^2} + B x + C y + D = 0$
provided that:
:$B^2 + C^2 \ge 4 A D$
:$A > 0$.
Thus:
By setting:
:$\alpha := A$, $\beta := \dfrac B 2 + \dfrac C {2 i}$ and $\gamma := D$
we have:
:$\alpha z \overline z + \beta z + \overline \beta \overline z + \gamma = 0$
As $A, B, C, D \in \R$ it follows that both $\alpha$ and $\gamma$ are real.
Then we have that $A > 0$ and so $\alpha > 0$.
Then note that:
Given that:
:$B^2 + C^2 \ge 4 A D$
it follows that:
:$\cmod \beta^2 > \alpha \gamma$
If $\alpha = 0$ and $\beta \ne 0$ the equation devolves to:
:$\beta z + \overline \beta \overline z + \gamma = 0$
which from Equation of Line in Complex Plane: Formulation 1 is the equation of a straight line.
The result follows.
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Complex Conjugate Coordinates: $44$
Category:Equation of Circle in Complex Plane
Equation of Circular Arc in Complex Plane

== Theorem ==
Let $a, b \in \C$ be complex constants representing the points $A$ and $B$ respectively in the complex plane.
Let $z \in \C$ be a complex variable representing the point $Z$ in the complex plane.
Let $\lambda \in \R$ be a real constant such that $-\pi < \lambda < \pi$.
Then the equation:
:$\arg \dfrac {z - b} {z - a} = \lambda$
represents the arc of a circle with $AB$ as a chord subtending an angle $\lambda$ at $Z$ on the circumference.
== Proof ==
:
By Geometrical Interpretation of Complex Subtraction:
:$z - a$ represents the line from $A$ to $Z$
:$z - b$ represents the line from $B$ to $Z$
Thus:
:$\arg \dfrac {z - b} {z - a} = \lambda$
represents the statement that the angle between $AZ$ and $BZ$ is constant:
:$\angle AZB = \lambda$
That is, the angle subtended by $AB$ at $Z$ is $\lambda$.
The result follows from the Inscribed Angle Theorem.
== Sources ==
* : $\S 2$. Geometrical Representations: Example $3$.
Category:Geometry of Complex Plane
Equation of Circular Helix/Parametric Form

== Theorem ==
Let $\HH$ be a circular helix embedded in Cartesian $3$-space whose axis coincides with the $z$-axis.
$\HH$ can be described by the parametric equation:
:$\begin {cases}
x & = a \cos t \\
y & = a \sin t \\
z & = b t \\
\end {cases}$
where $t$ is the parameter.
== Proof ==
== Sources ==
* 
* 
Category:Circular Helices
Equation of Cissoid of Diocles/Cartesian Form

== Theorem ==
The cissoid of Diocles can be defined by the Cartesian equation:
:$x \paren {x^2 + y^2} = 2 a y^2$
== Proof ==
== Also presented as ==
== Also see ==
* Equation of Cissoid of Diocles/Polar Form
* Equation of Cissoid of Diocles/Parametric Form
== Sources ==
* : $\S 11$: Special Plane Curves: Cissoid of Diocles: $11.33$
* 
* 
* 
* : $\S 9$: Special Plane Curves: Cissoid of Diocles: $9.33.$
* 
Category:Cissoid of Diocles
Equation of Cissoid of Diocles/Cartesian Form/Also presented as

== Equation of Cissoid of Diocles in Cartesian Form: Also presented as ==
The Cartesian equation for the cissoid of Diocles can also be defined as:
:$y^2 = \dfrac {x^3} {2 a - x}$
or:
:$y^2 \paren {2 a - x} = x^3$
which can be derived directly from the form given by subtracting $x y^2$ from both sides and rearranging.
== Sources ==
* : $\S 11$: Special Plane Curves: Cissoid of Diocles: $11.33$
* 
* 
* 
* : $\S 9$: Special Plane Curves: Cissoid of Diocles: $9.33.$
* 
Category:Cissoid of Diocles
Equation of Cissoid of Diocles/Parametric Form

== Theorem ==
The cissoid of Diocles can be defined by the parametric equation:
:$\begin {cases} x = 2 a \sin^2 \theta \\ \\ y = \dfrac {2 a \sin^3 \theta} {\cos \theta} \end {cases}$
== Proof ==
== Also see ==
* Equation of Cissoid of Diocles/Polar Form
* Equation of Cissoid of Diocles/Cartesian Form
== Sources ==
* : $\S 11$: Special Plane Curves: Cissoid of Diocles: $11.34$
* : $\S 9$: Special Plane Curves: Cissoid of Diocles: $9.34.$
* 
Category:Cissoid of Diocles
Equation of Cissoid of Diocles/Polar Form

== Theorem ==
The cissoid of Diocles can be defined by the polar equation:
:$r = 2 a \sin \theta \tan \theta$
== Proof ==
:
By construction:
== Also see ==
* Equation of Cissoid of Diocles/Cartesian Form
* Equation of Cissoid of Diocles/Parametric Form
== Sources ==
* 
* 
* 
Category:Cissoid of Diocles
Equation of Confocal Conics/Formulation 1

== Definition ==
The equation:
:$\dfrac {x^2} {a^2 + \lambda} + \dfrac {y^2} {b^2 + \lambda} = 1$
where:
:$\tuple {x, y}$ denotes an arbitrary point in the cartesian plane
:$a$ and $b$ are real constants such that $a^2 > b^2$
:$\lambda$ is a real parameter such that $a^2 > -\lambda$
defines the set of all confocal conics whose foci are at $\tuple {\pm \sqrt {a^2 - b^2}, 0}$.
== Proof ==
Let $b^2 > -\lambda$.
Then from Equation of Confocal Ellipses: Formulation 1, $(1)$ defines the set of all confocal ellipses whose foci are at $\tuple {\pm \sqrt {a^2 - b^2}, 0}$.
Let $b^2 < -\lambda < a^2$.
Then from Equation of Confocal Hyperbolas: Formulation 1, $(1)$ defines the set of all confocal hyperbolas whose foci are at $\tuple {\pm \sqrt {a^2 - b^2}, 0}$.
Hence the result.
== Also presented as ==
== Sources ==
* : Chapter $\text I$: Introductory: $\S 1.201$ The Differential Equation of a Family of Confocal Conics
Category:Equation of Confocal Conics
Equation of Confocal Conics/Formulation 1/Also presented as

== Equation of Confocal Conics: Formulation $1$: Also presented as ==
The equation for confocal conics can also be seen presented in the form:
:$\dfrac {x^2} {a^2 - k} + \dfrac {y^2} {b^2 - k} = 1$
where:
:$\tuple {x, y}$ denotes an arbitrary point in the cartesian plane
:$a$ and $b$ are real constants such that $a^2 > b^2$
:$k$ is a real parameter such that $a^2 > k$.
In this presentation:
:if $b^2 > k$, the equation generates confocal ellipses whose foci are at $\tuple {\pm \sqrt {a^2 - b^2}, 0}$
:if $b^2 < k < a^2$, the equation generates confocal hyperbolas whose foci are at $\tuple {\pm \sqrt {a^2 - b^2}, 0}$.
== Sources ==
* 
* 
Category:Equation of Confocal Conics
Equation of Confocal Conics/Formulation 2

== Definition ==
The equation:
:$(1): \quad \dfrac {x^2} {a^2} + \dfrac {y^2} {a^2 - c^2} = 1$
where:
:$\tuple {x, y}$ denotes an arbitrary point in the cartesian plane
:$c$ is a (strictly) positive constant
:$a$ is a (strictly) positive parameter
defines the set of all confocal conics whose foci are at $\tuple {\pm c, 0}$.
== Proof ==
Let $a > c$.
Then from Equation of Confocal Ellipses: Formulation 2, $(1)$ defines the set of all confocal ellipses whose foci are at $\tuple {\pm c, 0}$.
Let $a < c$.
Then from Equation of Confocal Hyperbolas, $(1)$ defines the set of all confocal hyperbolas whose foci are at $\tuple {\pm c, 0}$.
Hence the result.
== Also see ==
* Equation of Confocal Conics/Formulation 1
== Sources ==
* : $1$: The Nature of Differential Equations: Miscellaneous Problems for Chapter $1$: $6$
Category:Equation of Confocal Conics
Equation of Confocal Ellipses/Formulation 1

== Definition ==
The equation:
:$(1): \quad \dfrac {x^2} {a^2 + \lambda} + \dfrac {y^2} {b^2 + \lambda} = 1$
where:
:$\tuple {x, y}$ denotes an arbitrary point in the cartesian plane
:$a$ and $b$ are (strictly) positive constants such that $a^2 > b^2$
:$\lambda$ is a (strictly) positive parameter such that $b^2 > -\lambda$
defines the set of all confocal ellipses whose foci are at $\tuple {\pm \sqrt {a^2 - b^2}, 0}$.
== Proof ==
Let $a$ and $b$ be arbitrary (strictly) positive real numbers fulfilling the constraints as defined.
Let $E$ be the locus of the equation:
:$(1): \quad \dfrac {x^2} {a^2 + \lambda} + \dfrac {y^2} {b^2 + \lambda} = 1$
As $b^2 > -\lambda$ it follows that:
:$b^2 + \lambda > 0$
and as $a^2 > b^2$:
:$a^2 + \lambda > 0$
Thus $(1)$ is in the form:
:$\dfrac {x^2} {r^2} + \dfrac {y^2} {s^2} = 1$
where:
:$r^2 = a^2 + \lambda$
:$s^2 = b^2 + \lambda$
From Equation of Ellipse in Reduced Form, this is the equation of an ellipse in reduced form.
It follows that:
:$\tuple {\pm \sqrt {a^2 + \lambda}, 0}$ are the positions of the vertices of $E$
:$\tuple {0, \pm \sqrt {b^2 + \lambda} }$ are the positions of the covertices of $E$
From Focus of Ellipse from Major and Minor Axis, the positions of the foci of $E$ are given by:
:$\paren {a^2 + \lambda} - \paren {b^2 + \lambda} = c^2$
where $\tuple {\pm c, 0}$ are the positions of the foci of $E$.
Thus we have:
Hence the result.
== Also see ==
* Equation of Confocal Conics
* Equation of Confocal Hyperbolas
Category:Equation of Confocal Ellipses
Equation of Confocal Ellipses/Formulation 2

== Definition ==
The equation:
:$(1): \quad \dfrac {x^2} {a^2} + \dfrac {y^2} {a^2 - c^2} = 1$
where:
:$\tuple {x, y}$ denotes an arbitrary point in the cartesian plane
:$c$ is a (strictly) positive constant
:$a$ is a (strictly) positive parameter such that $a > c$
defines the set of all confocal ellipses whose foci are at $\tuple {\pm c, 0}$.
== Proof ==
Let $a$ and $c$ be arbitrary (strictly) positive real numbers fulfilling the constraints as defined.
Let $E$ be the locus of the equation:
:$(1): \quad \dfrac {x^2} {a^2} + \dfrac {y^2} {a^2 - c^2} = 1$
As $a > c$ it follows that:
:$a^2 > c^2$
and so:
:$a^2 - c^2 > 0$
Thus $(1)$ is in the form:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
From Equation of Ellipse in Reduced Form, this is the equation of an ellipse in reduced form.
Thus:
:$\tuple {\pm a, 0}$ are the positions of the vertices of $E$
:$\tuple {0, \pm b}$ are the positions of the covertices of $E$
From Focus of Ellipse from Major and Minor Axis:
:$\tuple {\pm c, 0}$ are the positions of the foci of $E$.
Hence the result.
== Also see ==
* Equation of Confocal Conics
* Equation of Confocal Hyperbolas
== Sources ==
* : $1$: The Nature of Differential Equations: Miscellaneous Problems for Chapter $1$: $6$
Category:Equation of Confocal Ellipses
Equation of Confocal Hyperbolas/Formulation 1

== Definition ==
The equation:
:$(1): \quad \dfrac {x^2} {a^2 + \lambda} + \dfrac {y^2} {b^2 + \lambda} = 1$
where:
:$\tuple {x, y}$ denotes an arbitrary point in the cartesian plane
:$a$ and $b$ are (strictly) positive constants such that $a^2 > b^2$
:$\lambda$ is a (strictly) positive parameter such that $b^2 < -\lambda < a^2$
defines the set of all confocal hyperbolas whose foci are at $\tuple {\pm \sqrt {a^2 + b^2}, 0}$.
== Proof ==
Let $a$ and $b$ be arbitrary (strictly) positive real numbers fulfilling the constraints as defined.
Let $E$ be the locus of the equation:
:$(1): \quad \dfrac {x^2} {a^2 + \lambda} + \dfrac {y^2} {b^2 + \lambda} = 1$
As $b^2 < -\lambda$ it follows that:
:$b^2 + \lambda < 0$
and as $-\lambda < a^2$:
:$a^2 + \lambda > 0$
Thus $(1)$ is in the form:
:$\dfrac {x^2} {r^2} - \dfrac {y^2} {s^2} = 1$
where:
:$r^2 = a^2 + \lambda$
:$s^2 = -\lambda + b^2$
From Equation of Hyperbola in Reduced Form, this is the equation of an hyperbola in reduced form.
It follows that:
:$\tuple {\pm \sqrt {a^2 + \lambda}, 0}$ are the positions of the vertices of $E$
:$\tuple {0, \pm \sqrt {b^2 - \lambda} }$ are the positions of the covertices of $E$
From Focus of Hyperbola from Transverse and Conjugate Axis, the positions of the foci of $E$ are given by:
:$\paren {a^2 + \lambda} + \paren {b^2 - \lambda} = c^2$
where $\tuple {\pm c, 0}$ are the positions of the foci of $E$.
Thus we have:
Hence the result.
== Also see ==
* Equation of Confocal Conics
* Equation of Confocal Ellipses
Category:Equation of Confocal Hyperbolas
Equation of Confocal Hyperbolas/Formulation 2

== Definition ==
The equation:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {a^2 - c^2} = 1$
where:
:$\tuple {x, y}$ denotes an arbitrary point in the cartesian plane
:$c$ is a (strictly) positive constant
:$a$ is a (strictly) positive parameter such that $a < c$
defines the set of all confocal hyperbolas whose foci are at $\tuple {\pm c, 0}$.
== Proof ==
Let $a$ and $c$ be arbitrary (strictly) positive real numbers fulfilling the constraints as defined.
Let $H$ be the locus of the equation:
:$(1): \quad \dfrac {x^2} {a^2} + \dfrac {y^2} {a^2 - c^2} = 1$
As $a < c$ it follows that:
:$a^2 < c^2$
and so:
:$a^2 - c^2 < 0$
Thus $(1)$ is in the form:
:$\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 1$
Thus from Equation of Hyperbola in Reduced Form, $H$ defines an hyperbola where:
:$\tuple {\pm a, 0}$ are the positions of the vertices of $H$
:the transverse axis of $H$ has length $2 a$
:the conjugate axis of $H$ has length $2 b$
From Focus of Hyperbola from Transverse and Conjugate Axis
:$\tuple {\pm c, 0}$ are the positions of the foci of $H$.
Hence the result.
== Also see ==
* Equation of Confocal Conics
* Equation of Confocal Ellipses
== Sources ==
* : $1$: The Nature of Differential Equations: : Miscellaneous Problems for Chapter $1$: $6$
Category:Equation of Confocal Hyperbolas
Equation of Conic Section/Cartesian Form

== Theorem ==
The general conic section can be expressed in Cartesian coordinates in the form:
:$a x^2 + b x y + c y^2 + d x + e y + f = 0$
for some $a, b, c, d, e, f \in \R$.
== Proof ==
By definition, a conic section is the set of points of intersection between a cone and a plane.
Let $P = \tuple {\alpha, \beta, \gamma}$ be the apex of the cone.
Let $Q = \tuple {x, y, z}$ be a point of intersection between the plane and the cone.
From Equation of Right Circular Cone, we have:
:$(1): \quad \paren {x - \alpha}^2 + \paren {y - \beta}^2 = \eta \paren {z - \gamma}^2$
for some (strictly) positive real number $\eta$.
From Equation of Plane, we have:
$A x + B y + C z + D = 0$
for some real numbers $A, B, C, D$.
Let $C \ne 0$.
Then:
:$z = - D - \dfrac {A x + B y} C$
Hence, plugging in $z$ into $(1)$:
:$\paren {x - \alpha}^2 + \paren {y - \beta}^2 = \eta \paren {\dfrac {A x + B y} C + D + \gamma}^2$
Opening up the brackets and making the corresponding notation of constants, we get the desired result.
Let $C = 0$.
Then:
:$A x + B y + D = 0$
In order for the plane to be well-defined, $A$ or $B$ must be non-zero.
, since the cone equation $(1)$ is symmetric in $x$ and $y$, we can consider $B \ne 0$ (otherwise interchange coordinates).
Then:
:$y = - D - \dfrac {A x} B$
Hence, plugging $y$ into $(1)$:
:$\paren {x - \alpha}^2 + \paren {-D - \dfrac {A x} B - \beta}^2 = \eta \paren {z - \gamma}^2$
Opening up the brackets and making the corresponding notation of constants and coordinates ($z \leftrightarrow y$), we get the desired result.
== Also presented as ==
== Sources ==
* : Chapter $6$: Curves and Coordinates: Cartesian coordinates
Category:Equation of Conic Section
Equation of Conic Section/Cartesian Form/Also presented as

== Equation of Conic Section: Cartesian Form: Also presented as ==
The equation for the general conic section can also be found expressed in the form:
:$a x^2 + 2 h x y + b y^2 + 2 g x + 2 f y + c = 0$
for some $a, b, c, f, g, h \in \R$.
== Sources ==
* 
* 
Category:Equation of Conic Section
Equation of Conic Section/Cartesian Form/Discriminant Form

== Theorem ==
Let $K$ be a conic section embedded in a Cartesian plane with the general equation:
:$a x^2 + 2 h x y + b y^2 + 2 g x + 2 f y + c = 0$
where $a, b, c, f, g, h \in \R$.
Then after translation of coordinate axes, $K$ can be described using the equation:
:$a x^2 + 2 h x y + b y^2 - \dfrac \Delta {h^2 - a b} = 0$
where $\Delta$ is the discriminant of $K$:
:$\delta = \begin {vmatrix} a & h & g \\ h & b & f \\ g & f & c \end {vmatrix}$
== Proof ==
== Sources ==
* 
* 
Category:Equation of Conic Section
Equation of Conic Section/Cartesian Form/Eccentricity

== Theorem ==
Let $K$ be a conic section embedded in a Cartesian plane such that:
:one focus of $K$ is at the origin
:the eccentricity of $K$ is $e$
:the directrix of $K$ is a distance $h$ from the origin.
Then $K$ can be described using the equation:
:$\paren {1 - e^2} x^2 + 2 e^2 h x + y^2 = e^2 h^2$
== Proof ==
== Sources ==
* 
* 
Category:Equation of Conic Section
Category:Eccentricity of Conic Section
Equation of Cornu Spiral/Parametric

== Theorem ==
Let $K$ be a Cornu spiral embedded in a Cartesian coordinate plane such that the origin coincides with the point at which $s = 0$.
Then $K$ can be expressed by the parametric equations:
:$\begin {cases} x = a \sqrt 2 \map {\operatorname C} {\dfrac s {a \sqrt 2} } \\ \\ y = a \sqrt 2 \map {\operatorname S} {\dfrac s {a \sqrt 2} } \end {cases}$
where:
:$\operatorname C$ denotes the Fresnel cosine integral function
:$\operatorname S$ denotes the Fresnel sine integral function.
== Proof ==
== Sources ==
* 
* 
Category:Cornu Spiral
Equation of Cycloid

== Theorem ==
Consider a circle of radius $a$ rolling without slipping along the x-axis of a cartesian plane.
Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis.
Consider the cycloid traced out by the point $P$.
Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane.
The point $P = \tuple {x, y}$ is described by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
== Proof ==
Let the circle have rolled so that the radius to the point $P = \tuple {x, y}$ is at angle $\theta$ to the vertical.
:
The center of the circle is at $\tuple {a \theta, a}$.
From the diagram above, we see:
:The $x$-coordinate is to the left of the center of the circle by $a \sin \theta$
:The $y$-coordinate is below the center of the circle by $a \cos \theta$
It follows that:
:$x = a \theta - a \sin \theta$
:$y = a - a \cos \theta$
whence the result.
== Also see ==
* Length of Arc of Cycloid
* Area under Arc of Cycloid
== Sources ==
* : $\S 11$: Special Plane Curves: Cycloid: $11.5$
* : $1$: The Nature of Differential Equations: $\S 6$: The Brachistochrone. Fermat and the Bernoullis
* : Chapter $\text {B}.21$: The Cycloid
* 
* 
* 
* : $\S 9$: Special Plane Curves: Cycloid: $9.5.$
* 
Category:Cycloids
Equation of Cycloid in Cartesian Coordinates

== Theorem ==
Consider a circle of radius $a$ rolling without slipping along the $x$-axis of a cartesian plane.
Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis.
Consider the cycloid traced out by the point $P$.
Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane.
The point $P = \tuple {x, y}$ is described by the equation:
:$a \sin^{-1} \paren {\dfrac {\sqrt {2 a y - y^2} } a} = \sqrt {2 a y - y^2} + x$
== Proof ==
From Equation of Cycloid, the point $P = \tuple {x, y}$ is described by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Expressing $\theta$ and $\sin \theta$ in terms of $y$:
Substituting for $\theta$ and $\sin \theta$ in the expression for $x$:
== Sources ==
* : Chapter $\text {B}.21$: The Cycloid: Problem $1$
Category:Cycloids
Equation of Deltoid

== Theorem ==
Let $H$ be the deltoid generated by the epicycle $C_1$ of radius $b$ rolling without slipping around the inside of a deferent $C_2$ of radius $a = 3 b$.
Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin.
Let $P$ be a point on the circumference of $C_1$.
Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \tuple {a, 0}$ on the $x$-axis.
Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane.
The point $P = \tuple {x, y}$ is described by the parametric equation:
:$\begin{cases}
x & = 2 b \cos \theta + b \cos 2 \theta \\
y & = 2 b \sin \theta - b \sin 2 \theta
\end{cases}$
where $\theta$ is the angle between the $x$-axis and the line joining the origin to the center of $C_1$.
== Proof ==
By definition, a deltoid is a hypocycloid with $3$ cusps.
:
By Equation of Hypocycloid, the equation of $H$ is given by:
:$\begin{cases}
x & = \paren {a - b} \cos \theta + b \map \cos {\paren {\dfrac {a - b} b} \theta} \\
y & = \paren {a - b} \sin \theta - b \map \sin {\paren {\dfrac {a - b} b} \theta}
\end{cases}$
From Number of Cusps of Hypocycloid from Integral Ratio of Circle Radii, this can be generated by a epicycle $C_1$ of radius $\dfrac 1 3$ the radius of the deferent.
Thus $a = 3 b$ and the equation of $H$ is now given by:
:$\begin{cases}
x & = 2 b \cos \theta + b \cos 2 \theta \\
y & = 2 b \sin \theta - b \sin 2 \theta
\end{cases}$
== Sources ==
* : Chapter $\text {B}.21$: The Cycloid: Problem $11$
Category:Deltoids (Hypocycloids)
Equation of Ellipse in Complex Plane

== Theorem ==
Let $\C$ be the complex plane.
Let $E$ be an ellipse in $\C$ whose major axis is $d \in \R_{>0}$ and whose foci are at $\alpha, \beta \in \C$.
Then $C$ may be written as:
:$\cmod {z - \alpha} + \cmod {z - \beta} = d$
where $\cmod {\, \cdot \,}$ denotes complex modulus.
=== Interior ===
=== Exterior ===
== Proof ==
By definition of complex modulus:
:$\cmod {z - \alpha}$ is the distance from $z$ to $\alpha$
:$\cmod {z - \beta}$ is the distance from $z$ to $\beta$.
Thus $\cmod {z - \alpha} + \cmod {z - \beta}$ is the sum of the distance from $z$ to $\alpha$ and from $z$ to $\beta$.
This is precisely the equidistance property of the ellipse.
From Equidistance of Ellipse equals Major Axis, the constant distance $d$ is equal to the major axis of $E$.
== Examples ==
Category:Equation of Ellipse in Complex Plane
Category:Ellipses
Category:Geometry of Complex Plane
Equation of Ellipse in Complex Plane/Examples

== Examples of Equation of Ellipse in Complex Plane ==
=== Example: Foci at $1$ and $i$, Major Axis $4$ ===
=== Example: Foci at $-3$ and $3$, Major Axis $10$ ===
=== Example: Foci at $-2 i$ and $2 i$, Major Axis $6$ ===
=== Example: Foci at $-2 i$ and $2 i$, Major Axis $6$ ===
=== Example: Foci at $\tuple {2, -3}$ and $\tuple {-2, 3}$, Major Axis $10$ ===
Category:Equation of Ellipse in Complex Plane
Equation of Ellipse in Complex Plane/Examples/Foci at (0, 2), (0, -2), Major Axis 10

== Example of Equation of Ellipse in Complex Plane ==
The ellipse in the complex plane whose major axis is of length $10$ and whose foci are at the points corresponding to $\tuple {0, 2}$ and $\tuple {0, -2}$ is given by the equation:
:$\cmod {z + 2 i} + \cmod {z - 2 i} = 10$
== Proof ==
From Equation of Ellipse in Complex Plane, the ellipse whose major axis is $d$ and whose foci are at the points corresponding to $\alpha$ and $\beta$ is given by:
:$\cmod {z - \alpha} + \cmod {z - \beta} = d$
The points $\tuple {0, 2}$ and $\tuple {0, -2}$ correspond to the imaginary numbers $2 i$ and $-2 i$ respectively.
The result follows.
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $72 \ \text {(b)}$
Category:Equation of Ellipse in Complex Plane
Equation of Ellipse in Complex Plane/Examples/Foci at (2, -3), (-2, 3), Major Axis 10

== Example of Equation of Ellipse in Complex Plane ==
The inequality:
:$\cmod {z + 2 - 3 i} + \cmod {z - 2 + 3 i} < 10$
defines the inside of the ellipse in the complex plane whose major axis is of length $10$ and whose foci are at the points corresponding to $\tuple {2, -3}$ and $\tuple {2, -3}$.
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $73 \ \text {(d)}$
Category:Equation of Ellipse in Complex Plane
Equation of Ellipse in Complex Plane/Examples/Foci at -2i, 2i, Major Axis 6

== Example of Equation of Ellipse in Complex Plane ==
The ellipse in the complex plane whose major axis is of length $6$ and whose foci are at the points corresponding to $-2 i$ and $2 i$ is given by the equation:
:$\cmod {z + 2 i} + \cmod {z - 2 i} = 6$
== Proof ==
From Equation of Ellipse in Complex Plane, the ellipse whose major axis is $d$ and whose foci are at the points corresponding to $\alpha$ and $\beta$ is given by:
:$\cmod {z - \alpha} + \cmod {z - \beta} = d$
:
The result follows.
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $71 \ \text {(b)}$
Category:Equation of Ellipse in Complex Plane