nyuuzyou/wikis huggingface dataset:
Equation of Ellipse in Complex Plane/Examples/Foci at -3, 3, 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 $-3$ and $3$ is given by the equation:
:$\cmod {z + 3} + \cmod {z - 3} = 10$
and also as:
:$\dfrac {x^2} {25} + \dfrac {y^2} {16} = 1$
== 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 rest of the result follows from Equation of Ellipse in Reduced Form: Cartesian Frame.
== Sources ==
* : $1$: Solved Problems: Graphical Representations of Complex Numbers. Vectors: $13 \ \text{(b)}$
* : $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $74$
Category:Equation of Ellipse in Complex Plane
Equation of Ellipse in Complex Plane/Examples/Foci at 1, i, Major Axis 4

== Example of Equation of Ellipse in Complex Plane ==
The ellipse in the complex plane whose major axis is of length $4$ and whose foci are at the points corresponding to $1$ and $i$ is given by the equation:
:$\cmod {z - 1} + \cmod {z - i} = 4$
== 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 ==
* : $\S 2$. Geometrical Representations: Exercise $3$.
Category:Equation of Ellipse in Complex Plane
Equation of Ellipse in Reduced Form

== Theorem ==
Let $K$ be an ellipse aligned in a cartesian plane in reduced form.
Let:
:the major axis of $K$ have length $2 a$
:the minor axis of $K$ have length $2 b$.
=== Cartesian Frame ===
=== Parametric Form ===
== Also see ==
* Equation of Parabola in Reduced Form
* Equation of Hyperbola in Reduced Form
Category:Ellipses
Category:Equation of Ellipse in Reduced Form
Equation of Ellipse in Reduced Form/Cartesian Frame

== Theorem ==
The equation of $K$ is:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* 
* 
* 
Category:Equation of Ellipse in Reduced Form
Equation of Ellipse in Reduced Form/Cartesian Frame/Parametric Form

== Theorem ==
The equation of $K$ in parametric form is:
where $\theta$ is the eccentric angle of the point $P = \tuple {x, y}$ $K$.
== Proof ==
Let the point $\tuple {x, y}$ satisfy the equations:
Then:
== Sources ==
* 
* 
* 
Category:Equation of Ellipse in Reduced Form
Equation of Ellipse in Reduced Form/Cartesian Frame/Proof 1

== Theorem ==
== Proof ==
:
By definition, the foci $F_1$ and $F_2$ of $K$ are located at $\tuple {-c, 0}$ and $\tuple {c, 0}$ respectively.
Let the vertices of $K$ be $V_1$ and $V_2$.
By definition, these are located at $\tuple {-a, 0}$ and $\tuple {a, 0}$.
Let the covertices of $K$ be $C_1$ and $C_2$.
By definition, these are located at $\tuple {0, -b}$ and $\tuple {0, b}$.
Let $P = \tuple {x, y}$ be an arbitrary point on the locus of $K$.
From the equidistance property of $K$ we have that:
:$F_1 P + F_2 P = d$
where $d$ is a constant for this particular ellipse.
From Equidistance of Ellipse equals Major Axis:
:$(1): \quad d = 2 a$
Also, from Focus of Ellipse from Major and Minor Axis:
:$(2): \quad a^2 - c^2 = b^2$
Then:
== Sources ==
* 
Category:Equation of Ellipse in Reduced Form
Equation of Ellipse in Reduced Form/Cartesian Frame/Proof 2

== Theorem ==
== Proof ==
:
Let $P$ be an arbitrary point in the plane.
Let $PM$ be dropped perpendicular to $V_1 V_2$.
Hence $M = \tuple {x, 0}$.
From Intersecting Chord Theorem for Conic Sections:
:$PM^2 = k V_1 M \times M V_2$
for some constant $k$.
Hence:
Putting $x = 0$, we find the points where $K$ crosses the $y$-axis are defined by:
:$y^2 = k a^2$
This gives us:
:$k a^2 = b^2$
and so:
Hence the result.
== Sources ==
* : Chapter $\text {IV}$. The Ellipse: $2$. To find the equation of the ellipse in its simplest form
Category:Equation of Ellipse in Reduced Form
Equation of Envelope of Family of Curves

== Theorem ==
Let $\FF$ be a one-parameter family of curves defined by a parameter $m$.
Let $E$ be the equation defining $\FF$.
Let $E'$ be the equation obtained by equating to zero the partial derivative of $E$ $m$.
The equation defining the envelope of $\FF$ can then be found by eliminating $m$ between $E$ and $E'$.
== Proof ==
== Examples ==
== Sources ==
* 
* 
Category:Equation of Envelope of Family of Curves
Category:Envelopes
Equation of Envelope of Family of Curves/Examples/y = 2mx + m^2

== Examples of Equation of Envelope of Family of Curves ==
Consider the family of curves $\FF$ embedded in the Cartesian plane defined by the equation $E$:
:$E: \quad y = 2 m x + m^2$
where $m$ is the parameter of $\FF$.
The envelope of $\FF$ is the parabola whose equation is:
:$y = -x^2$
== Proof ==
Setting $2 x + 2 m = 0$:
== Sources ==
* 
* 
Category:Examples of Equation of Envelope of Family of Curves
Equation of Epicycloid

== Theorem ==
Let a circle $C_1$ of radius $b$ roll without slipping around the outside of a circle $C_2$ of radius $a$.
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 $H$ be the epicycloid 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:
== Proof ==
:
Let $C_1$ have rolled so that the line $OC$ through the radii of $C_1$ and $C_2$ is at angle $\theta$ to the $x$-axis.
Let $C_1$ have turned through an angle $\phi$ to reach that point.
By definition of sine and cosine, $P = \tuple {x, y}$ is defined by:
The arc of $C_1$ between $P$ and $B$ is the same as the arc of $C_2$ between $A$ and $B$.
Thus by Arc Length of Sector:
:$a \theta = b \phi$
Thus:
:$\phi + \theta = \paren {\dfrac {a + b} b} \theta$
whence the result.
== Sources ==
* : $\S 11$: Special Plane Curves: Epicycloid: $11.18$
* : Chapter $\text {B}.21$: The Cycloid: Problem $12$
* 
* 
* : $\S 9$: Special Plane Curves: Epicycloid: $9.18.$
Category:Epicycloids
Equation of Epitrochoid

== Theorem ==
Let a circle $C_1$ of radius $b$ roll around the outside of another circle $C_2$ of radius $a$.
Consider a point $P$ on the line of a radius of $C_1$ at a distance $d$ from the center of $C_1$.
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 $H$ be the epitrochoid 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:
== Proof ==
:
Let $C_1$ have rolled so that the line $OC$ through the radii of $C_1$ and $C_2$ is at angle $\theta$ to the $x$-axis.
Let $C_1$ have turned through an angle $\phi$ to reach that point.
By definition of sine and cosine, $P = \tuple {x, y}$ is defined by:
The arc of $C_1$ between $P$ and $B$ is the same as the arc of $C_2$ between $A$ and $B$.
Thus by Arc Length of Sector:
:$a \theta = b \phi$
Thus:
:$\phi + \theta = \paren {\dfrac {a + b} b} \theta$
whence the result.
== Sources ==
* 
* 
Category:Epitrochoids
Equation of Harmonic Wave

== Theorem ==
Let $\phi$ be a harmonic wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$.
Then the disturbance of $\phi$ at point $x$ and time $t$ can be expressed using the equation:
:$\map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
$\phi$ can also be expressed in the following forms:
=== In Terms of Wavelength and Velocity ===
=== In Terms of Wavelength and Period ===
=== In Terms of Wavelength and Frequency ===
=== In Terms of Wave Number and Frequency ===
== Proof ==
From Wave Profile of Harmonic Wave:
:$\forall x \in \R: \paren {\map \phi x}_{t \mathop = 0} = a \cos \omega x$
From Equation of Wave with Constant Velocity:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
Hence the result.
== Sources ==
* : Chapter $\text {I}$: The Equation of Wave Motion: $\S 3$: $(3)$
Category:Equation of Harmonic Wave
Category:Harmonic Waves
Equation of Harmonic Wave/Wave Number and Frequency

== Theorem ==
Let $\phi$ be a harmonic wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$.
Then the disturbance of $\phi$ at point $x$ and time $t$ can be expressed using the equation:
:$\map \phi {x, t} = a \map \cos {2 \pi \paren {k x - \nu t} }$
where:
:$k$ is the wave number of $\phi$
:$\nu$ is the frequency of $\phi$.
== Proof ==
== Sources ==
* : Chapter $\text {I}$: The Equation of Wave Motion: $\S 3$: $(11)$
Category:Equation of Harmonic Wave
Equation of Harmonic Wave/Wavelength and Frequency

== Theorem ==
Let $\phi$ be a harmonic wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$.
Then the disturbance of $\phi$ at point $x$ and time $t$ can be expressed using the equation:
:$\map \phi {x, t} = a \map \cos {2 \pi \paren {\dfrac x \lambda - \nu t} }$
where:
:$\lambda$ is the wavelength of $\phi$
:$\nu$ is the frequency of $\phi$.
== Proof ==
== Sources ==
* : Chapter $\text {I}$: The Equation of Wave Motion: $\S 3$: $(9)$
Category:Equation of Harmonic Wave
Equation of Harmonic Wave/Wavelength and Period

== Theorem ==
Let $\phi$ be a harmonic wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$.
Then the disturbance of $\phi$ at point $x$ and time $t$ can be expressed using the equation:
:$\map \phi {x, t} = a \map \cos {2 \pi \paren {\dfrac x \lambda - \dfrac t \tau} }$
where:
:$\lambda$ is the wavelength of $\phi$
:$\tau$ is the period of $\phi$.
== Proof ==
== Sources ==
* : Chapter $\text {I}$: The Equation of Wave Motion: $\S 3$: $(8)$
Category:Equation of Harmonic Wave
Equation of Harmonic Wave/Wavelength and Velocity

== Theorem ==
Let $\phi$ be a harmonic wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$.
Then the disturbance of $\phi$ at point $x$ and time $t$ can be expressed using the equation:
:$\map \phi {x, t} = a \map \cos {\dfrac {2 \pi} \lambda \paren {x - c t} }$
where $\lambda$ is the wavelength of $\phi$
== Proof ==
== Sources ==
* : Chapter $\text {I}$: The Equation of Wave Motion: $\S 3$: $(4)$
Category:Equation of Harmonic Wave
Equation of Horizontal Line

== Theorem ==
Let $\LL$ be a horizontal line embedded in the Cartesian plane $\CC$.
Then the equation of $\LL$ can be given by:
:$y = b$
where $\tuple {0, b}$ is the point at which $\LL$ intersects the $y$-axis.
:
== Proof ==
From the Normal Form of Equation of Straight Line in Plane, a general straight line can be expressed in the form:
:$x \cos \alpha + y \sin \alpha = p$
where:
:$p$ is the length of a perpendicular $\PP$ from $\LL$ to the origin
:$\alpha$ is the angle made between $\PP$ and the $x$-axis.
As $\LL$ is horizontal, then by definition $\PP$ is vertical.
By definition, the vertical line through the origin is the $y$-axis itself.
Thus:
:$\alpha$ is a right angle, that is $\alpha = \dfrac \pi 2 = 90 \degrees$
:$p = b$
Hence the equation of $\LL$ becomes:
Hence the result.
== Also see ==
* Equation of Vertical Line
Category:Equations of Straight Lines in Plane
Equation of Hyperbola in Complex Plane

== Theorem ==
Let $\C$ be the complex plane.
Let $H$ be a hyperbola 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.
== 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 difference of the distance from $z$ to $\alpha$ and from $z$ to $\beta$.
This is precisely the equidistance property of the hyperbola.
From Equidistance of Hyperbola equals Transverse Axis, the constant distance $d$ is equal to the transverse axis of $H$.
== Examples ==
Category:Hyperbolas
Category:Geometry of Complex Plane
Category:Equation of Hyperbola in Complex Plane
Equation of Hyperbola in Complex Plane/Examples

== Examples of Equation of Hyperbola in Complex Plane ==
=== Example: Foci at $3$ and $-3$, Transverse Axis $4$ ===
=== Hyperbola Defined by $\map \Im {z^2} = 4$ ===
=== Hyperbola Defined by $\map \Re {z^2} > 1$ ===
Category:Equation of Hyperbola in Complex Plane
Equation of Hyperbola in Complex Plane/Examples/Foci at 3, -3, Transverse Axis 4

== Example of Equation of Hyperbola in Complex Plane ==
The hyperbola in the complex plane whose transverse axis is of length $4$ and whose foci are at the points corresponding to $-3$ and $3$ is given by the equation:
:$\cmod {z + 3} - \cmod {z - 3} = 4$
== Proof ==
From Equation of Hyperbola in Complex Plane, the hyperbola whose transverse 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: $71 \ \text {(c)}$
Category:Equation of Hyperbola in Complex Plane
Equation of Hyperbola in Complex Plane/Examples/Imaginary Part of z^2 = 4

== Example of Use of Equation of Hyperbola in Complex Plane ==
The equation:
:$\map \Im {z^2} = 4$
describes a hyperbola embedded in the complex plane.
== Proof ==
This is the equation of a rectangular hyperbola:
:
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $71 \ \text {(e)}$
Category:Equation of Hyperbola in Complex Plane
Equation of Hyperbola in Complex Plane/Examples/Real Part of z^2 greater than 1

== Example of Use of Equation of Hyperbola in Complex Plane ==
The inequality:
:$\map \Re {z^2} > 1$
describes the area shaded yellow defined by the following hyperbola:
:
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $73 \ \text {(b)}$
Category:Equation of Hyperbola in Complex Plane
Equation of Hyperbola in Reduced Form

== Theorem ==
Let $K$ be a hyperbola such that:
:the transverse axis of $K$ has length $2 a$
:the conjugate axis of $K$ has length $2 b$.
=== Cartesian Frame ===
=== Parametric Form 1 ===
=== Parametric Form 2 ===
=== Polar Frame ===
== Also see ==
* Equation of Parabola in Reduced Form
* Equation of Ellipse in Reduced Form
Category:Equation of Hyperbola in Reduced Form
Category:Hyperbolas
Equation of Hyperbola in Reduced Form/Cartesian Frame

== Theorem ==
Let $K$ be aligned in a cartesian plane in reduced form.
$K$ can be expressed by the equation:
:$\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 1$
=== Parametric Form 1 ===
=== Parametric Form 2 ===
== Proof 1 ==
== Proof 2 ==
== Sources ==
* 
* 
* 
Category:Equation of Hyperbola in Reduced Form
Equation of Hyperbola in Reduced Form/Cartesian Frame/Parametric Form 1

== Theorem ==
Let $K$ be aligned in a cartesian plane in reduced form.
The right-hand branch of $K$ can be expressed in parametric form as:
:$\begin {cases} x = a \cosh \theta \\ y = b \sinh \theta \end {cases}$
== Proof ==
Let the point $\tuple {x, y}$ satisfy the equations:
Then:
== Sources ==
* 
* 
Category:Equation of Hyperbola in Reduced Form
Equation of Hyperbola in Reduced Form/Cartesian Frame/Parametric Form 2

== Theorem ==
Let $K$ be aligned in a cartesian plane in reduced form.
$K$ can be expressed in parametric form as:
:$\begin {cases} x = a \sec \theta \\ y = b \tan \theta \end {cases}$
== Proof ==
Let the point $\tuple {x, y}$ satisfy the equations:
Then:
== Sources ==
* 
* 
* 
Category:Equation of Hyperbola in Reduced Form
Equation of Hyperbola in Reduced Form/Cartesian Frame/Proof 1

== Theorem ==
== Proof ==
 
:
By definition, the foci $F_1$ and $F_2$ of $K$ are located at $\tuple {-c, 0}$ and $\tuple {c, 0}$ respectively.
Let the vertices of $K$ be $V_1$ and $V_2$.
By definition, these are located at $\tuple {-a, 0}$ and $\tuple {a, 0}$.
Let the covertices of $K$ be $C_1$ and $C_2$.
By definition, these are located at $\tuple {0, -b}$ and $\tuple {0, b}$.
Let $P = \tuple {x, y}$ be an arbitrary point on the locus of $K$.
From the equidistance property of $K$ we have that:
:$\size {F_1 P - F_2 P} = d$
where $d$ is a constant for this particular ellipse.
From Equidistance of Hyperbola equals Transverse Axis:
:$d = 2 a$
Also, from Focus of Hyperbola from Transverse and Conjugate Axis:
:$c^2 a^2 = b^2$
, let us choose a point $P$ such that $F_1 P > F_2 P$.
Then:
== Sources ==
* 
Category:Equation of Hyperbola in Reduced Form
Equation of Hyperbola in Reduced Form/Polar Frame

== Theorem ==
Let $K$ be aligned in a polar plane in reduced form.
$K$ can be expressed by the equation:
:$\dfrac {\cos^2 \theta} {a^2} - \dfrac {\sin^2 \theta} {b^2} = \dfrac 1 {r^2}$
== Proof ==
Let the polar plane be aligned with its corresponding Cartesian plane in the conventional manner.
We have that
:$\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 1$
From Conversion between Cartesian and Polar Coordinates in Plane:
Hence:
== Sources ==
* : Chapter $\text {V}$. The Hyperbola: $3$. Asymptotes
Category:Equation of Hyperbola in Reduced Form
Equation of Hyperboloid of One Sheet

== Theorem ==
Let $\HH$ be a hyperboloid of one sheet.
Let $\HH$ be embedded in a cartesian $3$-space such that the conjugate axes of the hyperbolas forming its okane sections coincide with the $z$-axis.
The equation for $\HH$ can be expressed in the form:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} - \dfrac {z^2} {c^2} = 1$
where $a, b, c \in \R_{\ne 0}$.
== Proof ==
== Also see ==
* Equation of Hyperboloid of Two Sheets
== Sources ==
* : $\S 12$: Formulas from Solid Analytic Geometry: Hyperboloid of Two Sheets: $12.28$
* 
* 
* : $\S 10$: Formulas from Solid Analytic Geometry: Hyperboloid of Two Sheets: $10.28.$
Category:Hyperboloids of One Sheet
Equation of Hyperboloid of Two Sheets

== Theorem ==
Let $\HH$ be a hyperboloid of two sheets.
Let $\HH$ be embedded in a cartesian $3$-space such that the conjugate axes of the hyperbolas forming its okane sections coincide with the $x$-axis.
The equation for $\HH$ can be expressed in the form:
:$\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} - \dfrac {z^2} {c^2} = 1$
where $a, b, c \in \R_{\ne 0}$.
== Proof ==
== Also see ==
* Equation of Hyperboloid of One Sheet
== Sources ==
* : $\S 12$: Formulas from Solid Analytic Geometry: Hyperboloid of Two Sheets: $12.29$
* 
* 
* : $\S 10$: Formulas from Solid Analytic Geometry: Hyperboloid of Two Sheets: $10.29.$
Category:Hyperboloids of Two Sheets
Equation of Hypocycloid

== Theorem ==
Let a circle $C_1$ of radius $b$ roll without slipping around the inside of a circle $C_2$ of (larger) radius $a$.
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 $H$ be the hypocycloid 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:
:$x = \paren {a - b} \cos \theta + b \map \cos {\paren {\dfrac {a - b} b} \theta}$
:$y = \paren {a - b} \sin \theta - b \map \sin {\paren {\dfrac {a - b} b} \theta}$
== Proof ==
:
Let $C_1$ have rolled so that the line $OB$ through the radii of $C_1$ and $C_2$ is at angle $\theta$ to the $x$-axis.
Let $C_1$ have turned through an angle $\phi$ to reach that point.
By definition of sine and cosine, $P = \tuple {x, y}$ is defined by:
:$x = \paren {a - b} \cos \theta + b \map \cos {\phi - \theta}$
:$y = \paren {a - b} \sin \theta - b \map \sin {\phi - \theta}$
The arc of $C_1$ between $P$ and $B$ is the same as the arc of $C_2$ between $A$ and $B$.
Thus by Arc Length of Sector:
:$ a \theta = b \phi$
Thus:
:$\phi - \theta = \paren {\dfrac {a - b} b} \theta$
whence the result.
== Sources ==
* : $\S 11$: Special Plane Curves: Hypocycloid: $11.19$
* : Chapter $\text {B}.21$: The Cycloid
* : $\S 9$: Special Plane Curves: Hypocycloid: $9.19.$
Category:Hypocycloids
Equation of Imaginary Axis in Complex Plane

== Theorem ==
Let $\C$ be the complex plane.
Let $z \in \C$ be subject to the condition:
:$\cmod {z - 1} = \cmod {z + 1}$
where $\cmod {\, \cdot \,}$ denotes complex modulus.
Then the locus of $z$ is the imaginary axis.
== Proof ==
The result follows by definition of imaginary axis.
== Sources ==
* : $\S 2$. Geometrical Representations: Example $3$: $(2.5)$
Category:Geometry of Complex Plane
Equation of Limaçon of Pascal/Cartesian Form

== Theorem ==
The limaçon of Pascal can be defined by the Cartesian equation:
:$\paren {x^2 + y^2 - a x}^2 = b^2 \paren {x^2 + y^2}$
== Proof ==
== Also see ==
* Equation of Limaçon of Pascal/Polar Form
Category:Limaçons of Pascal
Equation of Limaçon of Pascal/Polar Form

== Theorem ==
The limaçon of Pascal can be defined by the polar equation:
:$r = b + a \cos \theta$
== Proof ==
Let $C$ be a circle of diameter $a$ whose circumference passes through the origin $O$.
Let the diameter of $C$ which passes through $O$ lie on the polar axis.
Let $OQ$ be a chord of $C$.
Let $b$ be a real constant.
:
Let $P = \polar {r, \theta}$ denote an arbitrary point on a limaçon of Pascal $L$.
We have that:
When $-\dfrac \pi 2 \le \theta \le \dfrac \pi 2$, we have:
:$r = b + a \cos \theta$
When $\dfrac \pi 2 \le \theta \le \dfrac {3 \pi} 2$, we have:
Hence the result.
== Also see ==
* Equation of Limaçon of Pascal/Cartesian Form
== Sources ==
* : $\S 11$: Special Plane Curves: Limacon of Pascal: $11.32$
* 
* 
* : $\S 9$: Special Plane Curves: Limacon of Pascal: $9.32.$
* 
Category:Limaçons of Pascal
Equation of Line in Complex Plane/Formulation 1

== Theorem ==
Let $\C$ be the complex plane.
Let $L$ be a straight line in $\C$.
Then $L$ may be written as:
:$\beta z + \overline \beta \overline z + \gamma = 0$
where $\gamma$ is real and $\beta$ may be complex.
== Proof ==
From Equation of Straight Line in Plane, the equation for a straight line is:
:$A x + B y + C = 0$
Thus:
The result follows by setting $\beta := \dfrac A 2 + \dfrac B {2 i}$ and $\gamma := C$.
== Sources ==
* : $\S 2$. Geometrical Representations: Exercise $5$.
* : $1$: Complex Numbers: Solved Problems: Complex Conjugate Coordinates: $44$
Category:Equation of Line in Complex Plane
Equation of Line in Complex Plane/Formulation 2

== Theorem ==
Let $\C$ be the complex plane.
Let $L$ be the infinite straight line in $\C$ which is the locus of the equation:
:$l x + m y = 1$
Then $L$ may be written as:
:$\map \Re {a z} = 1$
where $a$ is the point in $\C$ defined as:
:$a = l - i m$
== Proof ==
Let $z = x + i y$.
Let $a = l - i m$.
Then:
== Sources ==
* : $\S 2$. Geometrical Representations: Example $4$: $(2.6)$
Category:Equation of Line in Complex Plane