Conic Sections – Complete Study Notes

Everyday observation: the water’s top surface inside a drinking glass is a circle when the glass is upright and becomes an ellipse when the glass is tilted.
Key insight: A flat surface (plane) slicing through a solid of revolution (e.g.
a cylindrical glass or a cone) produces characteristic curves called conic sections.

Definition: Curves generated when a plane intersects a right circular cone.
Two identical cones placed tip-to-tip form the complete figure (two "nappes").
Fundamental parts of the cone
- Vertex: common point where all straight lines (generators) meet.
- Axis: line through the vertex dividing the cone symmetrically.
- Generator: any straight line lying entirely on the cone’s surface.
Classes
- Non-degenerate conics: Parabola, Ellipse (special case: Circle), Hyperbola.
- Degenerate conics: Point, Line, Two intersecting lines.

Two nappes (upper & lower) extend infinitely.
A plane’s tilt relative to the generators and axis dictates which curve appears.

Formation rule: plane is parallel to exactly one generator (same slant as cone’s side); intersects only one nappe.
Characteristic shape: "U"-curve opening indefinitely.
Analytic-geometry model (Cartesian):
$y^2 = 4px \quad \text{or} \quad x^2 = 4py$
Real-world analogies: satellite dishes, car headlights, projectiles’ ideal paths.

Formation rule: plane cuts through one nappe, is not parallel to any generator, and is oblique to the axis.
General equation (centered at origin, major axis horizontal):
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \; a>b
Planetary orbits (Kepler’s First Law) are elliptical.

Additional condition: the same plane becomes perpendicular to the cone’s axis ⇒ $a=b$ .
Standard equation: $x^2 + y^2 = r^2$
Appears in an upright glass of water, coins, wheels.

Formation rule: plane cuts both nappes and is parallel to two opposite generators.
Two separate branches opening away from each other.
Standard rectangular-hyperbola form (transverse axis on x-axis):
$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
Applications: navigation using hyperbolic radio lines, asymptotes in rational functions.

Plane meets only the vertex without touching cone’s sides.
Visual result: a single point.
Algebraic analogue: ellipse/parabola/hyperbola with zero radius or distance parameters.

Plane through vertex cuts through both nappes (not tangent).
Produces an "X" (pair of lines).
Appears algebraically as the factorization of a conic into two distinct linear factors.

Fig. 2 Parabola, Fig. 3a Ellipse, Fig. 3b Circle, Fig. 4 Hyperbola (non-degenerate).
Fig. 5 Point, Fig. 6 Line, Fig. 7 Two Intersecting Lines (degenerate).
Miscellaneous numeric slide decorations (e.g., 50 30 40 … 130) carry no conceptual meaning—ignore for study purposes.

Conic sections: parabola, ellipse, circle, hyperbola.
Degenerate forms: point, line, two intersecting lines.
Determining factor: orientation & position of the intersecting plane relative to cone’s generators and axis.

Bridges algebraic (analytic-geometry equations) and geometric (plane-cone intersection) viewpoints.
Lays groundwork for precalculus topics: quadratic functions, polar coordinates, eccentricity, reflective properties.
Real-life ties: optics (reflecting telescopes), architecture (arches), astronomy (orbits), engineering (cooling towers’ hyperbolic structure).

Parabola (axis horizontal): $x = ay^2 + by + c$ or translated $(y-k)^2 = 4p(x-h)$ .
Ellipse: \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1,\; a>b.
Circle: $(x-h)^2 + (y-k)^2 = r^2$ .
Hyperbola (horizontal transverse): $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ .

Practice sketching each conic from its defining plane condition.
Memorize orientation rules (parallel to one generator ⇒ parabola, etc.).
Work example problems converting between geometric definitions and algebraic equations.