3.5 Contents

• Universal Properties
• Circles
• Ellipses
• Parabolas
• Hyperbolas
• Rotations 🔧

3.5.1 Universal Properties

Two-dimensional subsets of a 3D (double) cone surface in which the shapes are determined by the intersection of a 2D plane

General Equation

$$A⋅x^2+B⋅x⋅y+C⋅y^2+D⋅x+E⋅y+F=0$$ All conic sections will be represented without rotation until the section on rotations. In other words, $B=0$.

Discriminant & Related Properties

$$\Delta=B^2-4⋅A⋅C$$ Assuming no other values:

• If $B=0$, the function is even either vertically or horizontally (not rotated)
• If $Δ \lt 0$, the function is an ellipse
• If $Δ \lt 0$, $B=0$, and $A \lt C$, the function is an ellipse with a horizontal major axis
• If $Δ \lt 0$, $B=0$, and $A \gt C$, the function is an ellipse with a vertical major axis
• If $Δ \lt 0$, $B=0$, and $A=C$, the function is a circle
• If $Δ \lt 0$, $A=C$, $B=D=E=0$, the function is a circle centered at the origin
• If $Δ=0$, the function is a parabola
• If $C=D=F=0$, the function is a vertical parabola with the vertex at the origin
• If $A=E=F=0$, the function is a horizontal parabola with the vertex at the origin
• If $D=E=0$, the function is either an ellipse or hyperbola centered at the origin
• If $Δ \gt 0$, the function is a hyperbola
• If $Δ \gt 0$ and $A+C=0$, the function is a rectangular hyperbola, meaning the asymptotes are perpendicular

Focus, Directrix & Eccentricity

A focus/foci is a point or set of points around which a curve is guided A Directrix is a fixed line perpendicular to the (major) axis of a function, which is determined by its focus and curvature Eccentricity is a function's deviation from being circular, and a constant ratio given by $e=c/a$; $$\text{eccentricity}=\frac{\text{distance from any point to the focus}}{\text{distance from any point to the directrix}}$$

• Two conic sections are congruent if the eccentricity of each are equal
• Circle eccentricities are always 0
• Ellipse eccentricities are always between 0 and 1
• Parabola eccentricities are always 1
• Hyperbola eccentricities are always between 1 and ∞
• Line eccentricities are always ∞

Degenerate Conics

Conics when either a 2D plane intercepts the vertex of a double cone, or the result of the general equation yields a non-function by real algebraic definition

No result	$A⋅x^2+A⋅y^2+1=0$
Point	$A⋅x^2+A⋅y^2=0$
Line	$D⋅x+E⋅y+F=0$
Intersecting lines	$x^2-y^2=0$
Parallel lines	$x^2-1=0$

3.5.2 Circles

Closed curves with all points equidistant to an internal point

Circumference	$2·\pi·r$
Area	$\pi·r^2$
Arc length	$\theta·r$
Sector area	$\theta·r^2/2$
Chord length (k)	$2·r·\sin\Big(\frac{\theta}{2}\Big)$
Segment area	$\frac{r^2}{2}·\Big(\theta-2·\sin\Big(\frac{\theta}{2}\Big)·\cos\Big(\frac{\theta}{2}\Big)\Big)$
Conic general equation	$A·(x^2+y^2)+D·x+E·y+F=0,A≠0$
Standard equation	${(x-x_∘)}^2+{(y-y_∘)}^2=r^2$
Conic-standard conversions	$x_∘=-D/(2·A)$ $y_∘=-E/(2·A)$ $r^2=(D^2+E^2-4·A·F)/(4·A^2)$
Focus Coordinates	$(x_∘,y_∘)$
Eccentricity	$0$
Directrix	None

Annulus

A ring formed by two concentric circles in which all features of circles apply with respect to differences involving two radii

Perimeter	$2·\pi·(R+r)$
Area	$\pi·(R^2-r^2)$
Sector Area	$\theta·(R^2-r^2)/2$

Deductive Logic for Area

Apply the area function of regular polygons, using the circumference as the perimeter and the radius as the apothem $$A=(2·\pi·r)·\frac{r}{2}$$

Proof of Arc Length

The arc length is a fraction of the circumference, therefore can be found by the ratio to it and its angle $$\frac{a}{2·\pi·r}=\frac{\theta}{2·\pi}$$

Proof of Sector Area

The sector area is a fraction of the circle area, therefore can be found by the ratio to it and its angle $$\frac{A_S}{\pi·r^2}=\frac{\theta}{2·\pi}$$

Proof of Chord Length

Use the radius and half the chord length to form a right triangle Use the sine function for the angle $$\sin\Big(\frac{\theta}{2}\Big)=\frac{k}{2·r}$$ Multiply by $2·r$

Proof of Segment Area

The segment area is the sector area minus the triangular area between the center and chord $$A_S=\frac{\theta·r^2}{2}-A_t$$ For the triangular area, use the chord length equation for the base and right angle definition with respect to the radius to find the height

$$k=2·r·\sin\Big(\frac{\theta}{2}\Big)$$ $$h=r·\cos\Big(\frac{\theta}{2}\Big)$$

Substitute the triangular area using these values for $(k·h)/2$ $$A_S=\frac{\theta·r^2}{2}-\frac{r^2}{2}·2·\sin\Big(\frac{\theta}{2}\Big)·\cos\Big(\frac{\theta}{2}\Big)$$ Factor $$A_S=\frac{r^2}{2}·\Big(\theta-2·\sin\Big(\frac{\theta}{2}\Big)·\cos\Big(\frac{\theta}{2}\Big)\Big)$$

Conic-Standard Conversion

Given the conic general equation with the properties for a circle, group the $x$ terms and $y$ terms, and isolate the constant $$A·x^2+D·x+A·y^2+E·y=-F$$ Divide by $A$ $$x^2+\frac{D}{A}·x+y^2+\frac{E}{A}·y=-\frac{F}{A}$$ Complete the square for the $x$ and $y$ terms $$x^2+\frac{D}{A}·x+\frac{D^2}{4·A^2}+y^2+\frac{E}{A}·y+\frac{E^2}{4·A^2}=-\frac{F}{A}+\frac{D^2}{4·A^2}+\frac{E^2}{4·A^2}$$ Factor $${\Big(x+\frac{D}{2·A}\Big)}^2+{\Big(y+\frac{E}{2·A}\Big)}^2=\frac{D^2+E^2-4·A·F}{4·A^2}$$ Equate the coefficients to the standard equation

3.5.3 Ellipses

Closed ovular curves whose points are the result of a constant sum between two internal points

Perimeter	$4·a\lt 2·\pi·a, a\gt b$
Area	$\pi·a·b$
Conic General Equation	$A·x^2+C·y^2+D·x+E·y+F=0,A·C>0$
Standard Equation	${(x-x_∘)}^2/a^2+{(y-y_∘)}^2/b^2=1$
Conic-Standard Conversions	$A=b^2$ $C=a^2$ $D=-2·b^2·x_∘$ $E=-2·a^2·y_∘$ $F=b^2·{x_∘}^2+a^2·{y_∘}^2-a^2·b^2$ $a^2=C$ $b^2=A$ $x_∘=-D/(2·A)$ $y_∘=-E/(2·C)$
Orientation	horizontal if $C>A \land a>b$ vertical if $A>C \land b>a$
Center Coordinates	$(x_∘,y_∘)$
Foci Coordinates	$f=\big(x_∘ \pm \sqrt{a^2-b^2}, y_∘\big),a>b$ $f=\left(x_∘,y_∘ \pm \sqrt{b^2-a^2}\right),b>a$
Eccentricity	$e=f/a=\sqrt{1-b^2/a^2},a>b$ $e=f/b=\sqrt{1-a^2/b^2},b>a$
Directrix	$x=\pm a/e,a>b$ $y=\pm b/e,b>a$

Conic-Standard Conversion

Given the standard equation $$\frac{{(x-x_∘)}^2}{a^2}+\frac{{(y-y_∘)}^2}{b^2}=1$$ Multiply by $a^2·b^2$ $$b^2·{(x-x_∘)}^2+a^2·{(y-y_∘)}^2=a^2·b^2$$ Expand $$b^2·x^2-2·b^2·x·x_∘+b^2·x_∘^2+a^2·y^2-2·a^2·y·y_∘+a^2·y_∘^2=a^2·b^2$$ Rearrange to appear as the conic general equation $$b^2·x^2+a^2·y^2-2·b^2·x_∘·x-2·a^2·y_∘·y+b^2·x_∘^2+a^2·y_∘^2-a^2·b^2=0$$ Equate the coefficients to the conic equation

3.5.4 Parabolas

Open mirrored curves whose points are the same distance between a common internal point and an exterenal line.

Conic General Equations	vertical: $A·x^2+D·x+E·y+F=0$ horizontal: $C·y^2+D·x+E·y+F=0$
The following are in vertical form
Standard Equation	$y=a·x^2+b·x+y_∘$
Vertex Equation	$y=a·(x-x_∘)^2+y_∘$
Intercept Equation	$y=a·(x-x_1)(x-x_2)$
Discriminant	$\Delta=b^2-4·a·y_∘$ If $Δ \lt 0$, two $x$-intercepts If $Δ=0$, one $x$-intercept If $Δ>0$, no $x$-intercepts
Conic-Standard Conversion	$a=-A/E$ $b=-D/E$ $y_∘=-F/E$
Vertex Coordinates	$(x_∘,y_∘)=\big(-\frac{b}{2·a},y_∘-\frac{b^2}{4·a}\big)$
$x$-Intercepts	$\lbrace (x_1,0),(x_2,0) \rbrace = \big(\frac{-b \pm \sqrt{b^2-4·a·y_∘}}{2·a},0\big)$
Fucus Length from Vertex	$f=\frac{1}{4·a}$
Focus Coordinates	$\big(-\frac{b}{2·a},y_∘-\frac{b^2+1}{4·a}\big)$
Eccentricity	$1$
Drectrix	$y=-f$

Conic-Standard Conversion

Rearrange the conic general equation to isolate the $y$ term $$E·y=-A·x^2-D·x-F$$ Divide by $E$ $$y=-\frac{A}{E}·x^2-\frac{D}{E}·x-\frac{F}{E}$$ Equate the coefficients to the standard equation

Standard-Vertex Conversion

Given the standard equation, isolate the $x$ terms $$y-y_∘=a·x^2+b·x$$ Divide by $a$ $$\frac{y-y_∘}{a}=x^2+\frac{b·x}{a}$$ Complete the square $$\frac{y-y_∘}{a}+\frac{b^2}{4·a^2}=x^2+\frac{b·x}{a}+\frac{b^2}{4·a^2}$$ Factor $$\frac{y-y_∘}{a}+\frac{b^2}{4·a^2}={\bigg(x+\frac{b}{2·a}\bigg)}^2$$ Multiply by $a$ $$y-y_∘+\frac{b^2}{4·a}=a·{\bigg(x+\frac{b}{2·a}\bigg)}^2$$ Isolate $y$ $$y=a·{\bigg(x+\frac{b}{2·a}\bigg)}^2-\frac{b^2}{4·a}+y_∘$$ Equate the coefficients to the vertex equation

Standard-Intercept Conversion

Given the quadratic formula with $y=0$, find the zeros of $x$

3.5.5 Hyperbolas

A mirrored set of open mirrored curvers whose points are the difference between two common internal points.

Conic General Equation	$A·x^2+C·y^2+D·x+E·y+F=0, A·C \lt 0$
Standard Equation	$(x-x_∘)^2/a^2+(y-y_∘)^2/b^2=1, a^2·b^2 \lt 0$
Standard Equation (Real Terms Only)	$\pm (x-x_∘)^2/a^2 \mp (y-y_∘)^2/b^2=1$
Conic-Standard Conversion	$A=b^2$ $C=a^2$ $D=-2·b^2·x_∘$ $E=-2·a^2·y_∘$ $F=b^2·{x_∘}^2+a^2·{y_∘}^2-a^2·b^2$ $a^2=C$ $b^2=A$ $x_∘=-D/(2·A)$ $y_∘=-E/(2·C)$
Orientation	horizontal if $C \lt 0; a \isin ℂ$ vertical if $A \lt 0; b \isin ℂ$
Vertices	$(x_∘ \pm a,y_∘) \lor (x_∘,y_∘ \pm b)$
Focus Coordinates	$(x_∘ \pm \sqrt{a^2+b^2},y_∘) \lor (x_∘,y_∘ \pm \sqrt{a^2+b^2})$
Eccentricity	$e=f/a=\sqrt{1+b^2/a^2}$
Directrix	$x=\pm a^2/f \lor y=\pm b^2/f$
Asymptotes	$y= \pm b·(x-x_∘)/a+y_∘$

Conic-Standard Conversion

The proof is the same for ellipses, however since by definition $A·C \lt 0$, either $a$ or $b$ must be imaginary. For the real-numbers-only representation, one of the terms is negated for $a$ and $b$ to both be positive.

3.5.6 Rotations 🔧

Conic section rotations occur exclusively when in the general equation $B \neq 0$ $$A·x^2+B·x·y+C·y^2+D·x+E·y+F=0$$ The angle of rotation is used to identify a new axis The original components are used to determine the new axis with respect to the angle of rotation



$$x=x'·\cos(\theta)-y'·\sin(\theta)$$ $$y=x'·\sin(\theta)+y'·\cos(\theta)$$ $$A·\big( \big)^2+B·\big( \big)·\big( \big)+C·\big( \big)^2+D·\big( \big)+E·\big( \big)+F=0$$