Where does conic section come from?

Objectives

• By the end of the lesson you should be able to:
  • Define all four classical conic sections: circle, parabola, ellipse, hyperbola.
  • Identify/label parts of each conic (center, vertex, focus/foci, directrix, axis/axes, latus rectum, etc.).
  • Illustrate every conic, including the four primary types and the degenerate cases (point, single line, intersecting lines).

Motivation / Visual Warm-Up

• Slides show a series of coordinate-plane pictures that look like
  • Perfect round curve → Circle
  • U-shaped curve → Parabola
  • Flattened oval → Ellipse
  • Two mirrored “open” curves → Hyperbola
• Prompt: “What do you notice?” → students expected to see different shapes, symmetry, and relative “narrowness.”

Where Do Conic Sections Come From?

• Start with a double-napped cone (two identical cones placed tip-to-tip).
• Intersect the cone(s) with a single plane → the curve of intersection is a conic.
  • Orientation / position of the plane determines the type.
• Degenerate conics occur when the cutting plane crosses only through the vertex or tangentially.

Classification of Conics

• Circle – plane perpendicular to the cone axis and cutting only one nappe.
• Ellipse – plane angled so that it intersects one nappe entirely, but not parallel to any generator.
• Parabola – plane parallel to exactly one generator (slanted side) of the cone.
• Hyperbola – plane intersects both nappes and is more “vertical” than the generator (parallel to cone axis).

Degenerate Conics

• Point
  • Plane passes through the vertex and nowhere else.
• Single line
  • Plane is tangent to the cone along one generator.
• Pair of intersecting lines
  • Plane goes through the vertex and cuts both nappes.

Circle

• Definition: set of all points equidistant from a fixed point (the center).
• Notation / parts:
  • Center $C(h,k)$
  • Radius $r$
  • Any point on the circle $P(x,y)$
• Standard (center–radius) form
  $r^2 = (x-h)^2 + (y-k)^2$
• Symmetry: simultaneously symmetric about both axes through the center.

Parabola

• Definition: locus of points equidistant from a fixed point (focus $F$) and a fixed line (directrix $D$).
• Key parts/terminology:
  • Vertex $V$ – midpoint of perpendicular segment connecting focus to directrix.
  • Axis of symmetry – line through $F$ and $V$; reflects the curve onto itself.
  • Latus rectum – segment through $F$ perpendicular to the axis; endpoints lie on the parabola.
  • $c$ – distance from vertex to focus; also from vertex to directrix.
  • Eccentricity $e$ for a parabola is always $e=1$.
• Openings (4 possible): right, left, up, down → leads to four standard forms.
  • Right-opening: $(y-k)^2 = 4c(x-h)$ with c>0
  • Left-opening: $(y-k)^2 = 4c(x-h)$ with c<0
  • Upward-opening: $(x-h)^2 = 4c(y-k)$ with c>0
  • Downward-opening: $(x-h)^2 = 4c(y-k)$ with c<0
• Endpoints of the latus rectum are labeled $E<em>1,E</em>2$.

Ellipse

• Definition (distance-sum form): locus of all points whose sum of distances from two fixed points (foci $F<em>1,F</em>2$) is constant.
• Alternative definition (focus–directrix ratio): set of points where $\dfrac{\text{distance to focus}}{\text{distance to directrix}} = e$ with 0<e<1.
• Parts & notation:
  • Center $C(h,k)$ – intersection of major & minor axes.
  • Major (transverse) axis – line joining the vertices & foci; length $2a$.
  • Minor (conjugate) axis – perpendicular bisector of major; length $2b$.
  • Vertices $V<em>1,V</em>2$ – endpoints of major axis (distance $a$ from center).
  • Co-vertices $B<em>1,B</em>2$ – endpoints of minor axis (distance $b$ from center).
  • Foci are located along the major axis, distance $c$ from center where $c^2 = a^2 - b^2$.
  • Latus rectum (latera recta) – chord through each focus perpendicular to major axis; classical length $\dfrac{2b^2}{a}$ (not explicitly in slide, but implied “length of the latus rectum is 2 …”).
  • Eccentricity $e=\dfrac{c}{a}$ with 0<e<1. As $e\to1^-$, ellipse gets narrower.
• Standard forms (two orientations):
  • Horizontal major axis (major along $x$):
    $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{a^2-c^2} = 1$ where $b^2 = a^2 - c^2$.
  • Vertical major axis (major along $y$):
    $\frac{(x-h)^2}{a^2-c^2} + \frac{(y-k)^2}{a^2} = 1$ where $b^2 = a^2 - c^2$.
• Properties recap:
  • Length major axis $=2a$, minor axis $=2b$.
  • Endpoints of major = vertices; endpoints of minor = co-vertices.
  • Center is axes intersection.
  • 0<e<1.
  • When $c=0$ (i.e.
    $a=b$) the ellipse becomes a circle.

Hyperbola

• Definition (distance-difference form): locus of all points for which the absolute difference of distances to two fixed points (foci) is constant.
• Visual: resembles two mirrored parabolas opening in opposite directions.
• Formation: plane cuts both nappes, orientation parallel to the cone axis → regular hyperbola.
• Parts & notation:
  • Center – midpoint of segment joining vertices;
  • Vertices $V<em>1,V</em>2$ – each branch’s “closest” point to center; distance $a$ from center.
  • Foci $F<em>1,F</em>2$ – inside each branch; distance $c$ from center.
  • Transverse axis (major axis) – line through vertices, foci, and center.
  • Conjugate axis – line through center perpendicular to transverse axis; length $2b$.
  • Asymptotes – two lines the branches approach as $|x|$ or $|y|\to\infty$. Equation forms derived from box: slope $\pm\dfrac{b}{a}$ (horizontal) or $\pm\dfrac{a}{b}$ (vertical case).
  • b – perpendicular distance from vertex to each asymptote.
• Standard forms (two orientations):
  • Horizontal transverse axis:
    $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{c^2-a^2} = 1$
  • Vertical transverse axis:
    $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{c^2-a^2} = 1$
  • Relation among parameters: $c^2 = a^2 + b^2$.
• Eccentricity e = \dfrac{c}{a} > 1 (though slide mostly cited qualitative description).

Summary of Standard Forms (Quick Reference)

• Circle: $r^2 = (x-h)^2 + (y-k)^2$
• Parabola (four directions):
  • Right: (y-k)^2 = 4c(x-h),\; c>0
  • Left: (y-k)^2 = 4c(x-h),\; c<0
  • Up: (x-h)^2 = 4c(y-k),\; c>0
  • Down: (x-h)^2 = 4c(y-k),\; c<0
• Ellipse:
  • Horizontal major: $\dfrac{(x-h)^2}{a^2} + \dfrac{(y-k)^2}{b^2} = 1\quad (b^2=a^2-c^2)$
  • Vertical major: $\dfrac{(x-h)^2}{b^2} + \dfrac{(y-k)^2}{a^2} = 1\quad (b^2=a^2-c^2)$
• Hyperbola:
  • Horizontal: $\dfrac{(x-h)^2}{a^2} - \dfrac{(y-k)^2}{b^2} = 1\quad (c^2=a^2+b^2)$
  • Vertical: $\dfrac{(y-k)^2}{a^2} - \dfrac{(x-h)^2}{b^2} = 1\quad (c^2=a^2+b^2)$

Degenerate Cases – Graphical Behavior

• Circle degenerates to a point when $r=0$.
• Ellipse degenerates to:
  • Point (when $a=b=0$) – special circle case.
  • Line segment between the foci when $e\to1$ and the transverse axis collapses.
• Parabola degenerates to a single line when plane is tangent to the cone.
• Hyperbola degenerates to intersecting lines when plane passes through the cone’s vertex and intersects both nappes.

Connections & Additional Notes

• All conics can be viewed through the focus–directrix property with a single unifying parameter: eccentricity $e$.
  • $e=0$ → circle
  • 0<e<1 → ellipse
  • $e=1$ → parabola
  • e>1 → hyperbola
• Real-world relevance & examples (implied by imagery):
  • Circles: wheels, orbits (≈ circular).
  • Parabolas: satellite dishes, headlights (parallel light → focus).
  • Ellipses: planetary orbits (Kepler’s 1st law).
  • Hyperbolas: navigation (difference of distances – e.g., trilateration), cooling towers, radio antennas.
• Ethical / philosophical implication (classroom angle): understanding natural shapes enables engineering advances, but also requires responsibility (e.g., weapon guidance uses parabolic mirrors, hyperbolic navigation).