00.10.01 · precalc / conics

Conic sections (parabola, ellipse, hyperbola)

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Anchor (Master): Apollonius ~200 BC Conics; Kepler 1609 Astronomia Nova; projective classification via the discriminant

Intuition [Beginner]

Imagine slicing through a cone (think of an ice-cream cone) with a flat blade. If you cut straight across, perpendicular to the central axis, the slice is a circle. Tilt the blade a little and the slice stretches into an oval shape called an ellipse. Tilt the blade until it is parallel to the side of the cone and the curve opens up forever — a parabola. Tilt further and the slice cuts both halves of a double cone, giving two separate curves that face away from each other — a hyperbola.

These four curves — circle, ellipse, parabola, hyperbola — are the conic sections, and every one of them comes from intersecting a plane with a cone. The reason they matter is that nature uses them constantly: planets follow ellipses around the sun, thrown balls trace parabolas, and telescope mirrors are shaped from hyperbolic surfaces. The conic sections exist because slicing a cone at different angles produces fundamentally different curves, and the angle of the slice determines which curve you get.

Visual [Beginner]

A double cone drawn as two ice-cream cones joined at their tips, with four cutting planes at different angles. One horizontal slice produces a circle, one tilted slice produces an ellipse, one slice parallel to the side produces a parabola, and one steep slice cutting both halves produces a hyperbola.

The four curves look different but share a common origin. Each one is determined by a single parameter — how steeply the plane meets the cone.

Worked example [Beginner]

A parabola has its vertex at the origin and opens to the right with focal length $p = 2$ . Find its equation and the focus.

Step 1. A parabola opening to the right with vertex at $(0, 0)$ has equation $y^{2} = 4 p x$ .

Step 2. Substitute $p = 2$ : $y^{2} = 4 \cdot 2 \cdot x = 8 x$ .

Step 3. The focus is at $(p, 0) = (2, 0)$ .

What this tells us: once we know which way the parabola opens and its focal length, the equation and the focus are determined directly.

Check your understanding [Beginner]

Formal definition [Intermediate+]

A conic section is the set of points $P$ in the plane such that the ratio of the distance from $P$ to a fixed point $F$ (the focus) to the distance from $P$ to a fixed line $ℓ$ (the directrix) is a constant $e$ (the eccentricity):

\frac{d ( P , F )}{d ( P , ℓ )} = e .

The three types are classified by the value of $e$ :

Ellipse ( $0 \leq e < 1$ ). The standard form with centre at the origin and semi-axes $a \geq b > 0$ is
$\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, c^{2} = a^{2} - b^{2},$
where $c$ is the distance from the centre to each focus. The eccentricity is $e = c / a$ ^{[Lang — Basic Mathematics Ch. 9]}.
Parabola ( $e = 1$ ). The standard form with vertex at the origin and axis along the $x$ -axis is
$y^{2} = 4 p x,$
where $p > 0$ is the focal length (distance from vertex to focus). The focus is at $(p, 0)$ and the directrix is $x = - p$ .
Hyperbola ( $e > 1$ ). The standard form with centre at the origin is
$\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1, c^{2} = a^{2} + b^{2},$
with foci at $(\pm c, 0)$ and eccentricity $e = c / a$ . The asymptotes are $y = \pm (b / a) x$ .

General conic. Every conic section in $R^{2}$ is the zero set of a degree-two polynomial

A x^{2} + B x y + C y^{2} + D x + E y + F = 0

with $(A, B, C) \neq = (0, 0, 0)$ . The converse also holds: every such equation defines a conic (possibly degenerate — a point, a line pair, or empty).

Counterexamples to common slips

A circle is an ellipse with $e = 0$ (both foci coincide at the centre), but an ellipse with $e > 0$ is not a circle. Setting $a = b$ in the ellipse equation gives $x^{2} + y^{2} = a^{2}$ , the circle of radius $a$ .
The parabola $y^{2} = 4 p x$ opens to the right when $p > 0$ and to the left when $p < 0$ . A common error is to assume the parabola always opens upward; the orientation depends on which variable is squared.
A hyperbola has two separate branches. The equation $x^{2} / a^{2} - y^{2} / b^{2} = 1$ defines the left and right branches (where $∣ x ∣ \geq a$ ), not the upper and lower branches.

Key theorem with proof [Intermediate+]

Theorem (eccentricity classification). Let $C$ be the locus of points $P = (x, y)$ satisfying $d (P, F) / d (P, ℓ) = e$ for a fixed point $F$ and a fixed line $ℓ$ with $F \in / ℓ$ . Then:

If $0 \leq e < 1$ , $C$ is an ellipse.
If $e = 1$ , $C$ is a parabola.
If $e > 1$ , $C$ is a hyperbola.

Proof. Place the focus at $F = (c, 0)$ and the directrix as $x = c / e^{2}$ (a convenient normalisation). A point $P = (x, y)$ satisfies

\frac{( x - c ) ^{2} + y ^{2}}{∣ x - c / e ^{2} ∣} = e .

Square both sides:

(x - c)^{2} + y^{2} = e^{2} (x - \frac{c}{e ^{2}})^{2} .

Expand the right-hand side:

(x - c)^{2} + y^{2} = e^{2} x^{2} - 2 c x + \frac{c ^{2}}{e ^{2}} .

Expand the left:

x^{2} - 2 c x + c^{2} + y^{2} = e^{2} x^{2} - 2 c x + \frac{c ^{2}}{e ^{2}} .

Cancel $- 2 c x$ from both sides and collect:

(1 - e^{2}) x^{2} + y^{2} = \frac{c ^{2}}{e ^{2}} - c^{2} = c^{2} (\frac{1}{e ^{2}} - 1) = c^{2} \cdot \frac{1 - e ^{2}}{e ^{2}} .

Case 1: $e \neq = 1$ . Divide both sides by $c^{2} (1 - e^{2}) / e^{2}$ :

\frac{e ^{2} x ^{2}}{c ^{2}} + \frac{y ^{2}}{c ^{2} ( 1 - e ^{2} ) / e ^{2}} = 1.

For $e < 1$ this is $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ with $a = c / e$ and $b^{2} = c^{2} (1 - e^{2}) / e^{2} > 0$ : an ellipse.

For $e > 1$ the coefficient $(1 - e^{2})$ is negative, so the $y^{2}$ term is negative: this gives $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ with $a = c / e$ and $b^{2} = c^{2} (e^{2} - 1) / e^{2} > 0$ : a hyperbola.

Case 2: $e = 1$ . The equation $(1 - 1) x^{2} + y^{2} = 0$ reduces to $y^{2} = 0$ , which is degenerate under this particular coordinate choice. Re-deriving with the focus at $(p, 0)$ and directrix $x = - p$ gives $d (P, F) = d (P, ℓ)$ , hence $(x - p)^{2} + y^{2} = ∣ x + p ∣$ . Squaring: $(x - p)^{2} + y^{2} = (x + p)^{2}$ , yielding $y^{2} = 4 p x$ , a parabola. $□$

Bridge. The eccentricity classification builds toward 00.11.01 where the polar form $r = l / (1 + e cos θ)$ unifies all three conic types into a single equation parametrised by $e$ , and appears again in 00.11.02 where conic parametrisations give rational maps onto each type. The foundational reason the classification works is that the focus-directrix condition is a single quadratic relation in $x$ and $y$ , and the discriminant of the resulting general conic determines the type. The central insight is that the three conic types are not separate families but a single family indexed by a continuous parameter $e$ , and this is exactly the content that identifies the geometric slicing picture with the algebraic discriminant classification. Putting these together, the bridge is between the geometric definition (distance ratio) and the algebraic equation (degree-two curve), and the pattern generalises to the projective classification where all non-degenerate conics are projectively equivalent over the complex numbers.

Exercises [Intermediate+]

Exercise 7 (hard, symbolic).

Prove that the sum of the distances from any point on the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ to the two foci $(\pm c, 0)$ is the constant $2 a$ , where $c^{2} = a^{2} - b^{2}$ .

Hint

Let $P = (x, y)$ be on the ellipse. Write $d_{1} = d (P, F_{1})$ and $d_{2} = d (P, F_{2})$ using the distance formula, square each, and add.

Answer

Let $P = (x, y)$ satisfy $x^{2} / a^{2} + y^{2} / b^{2} = 1$ , so $y^{2} = b^{2} (1 - x^{2} / a^{2})$ . The distances are $d_{1} = (x - c)^{2} + y^{2}$ and $d_{2} = (x + c)^{2} + y^{2}$ . Compute:

$d_{1}^{2} = (x - c)^{2} + y^{2} = x^{2} - 2 c x + c^{2} + b^{2} - \frac{b ^{2} x ^{2}}{a ^{2}} = x^{2} (1 - \frac{b ^{2}}{a ^{2}}) - 2 c x + c^{2} + b^{2} .$

Since $1 - b^{2} / a^{2} = c^{2} / a^{2}$ , this becomes $\frac{c ^{2} x ^{2}}{a ^{2}} - 2 c x + a^{2} = (\frac{c x}{a} - a)^{2}$ .

Therefore $d_{1} = ∣ a - c x / a ∣ = a - c x / a$ (for $∣ x ∣ \leq a$ and $c < a$ , the expression $a - c x / a$ is positive). Similarly $d_{2} = a + c x / a$ . Adding: $d_{1} + d_{2} = 2 a$ .

Rubric: full credit for computing both squared distances, simplifying using $c^{2} = a^{2} - b^{2}$ , and adding.

Advanced results [Master]

Theorem 1 (general conic discriminant). The conic $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ with $(A, B, C) \neq = (0, 0, 0)$ is classified by the discriminant $Δ = B^{2} - 4 A C$ : ellipse if $Δ < 0$ (and non-degenerate), parabola if $Δ = 0$ , hyperbola if $Δ > 0$ .

Theorem 2 (focus-directrix equivalence). Every non-degenerate conic section has a focus and a directrix such that the locus satisfies $d (P, F) = e \cdot d (P, ℓ)$ , and conversely every such locus is a non-degenerate conic. The parameter $e$ is the eccentricity.

Theorem 3 (reflection property). For an ellipse, a ray emanating from one focus reflects off the curve to pass through the other focus. For a parabola, a ray from the focus reflects parallel to the axis. For a hyperbola, a ray toward one focus reflects as though coming from the other focus.

Theorem 4 (Dandelin spheres). Inside the cone, one can inscribe spheres tangent to both the cone and the cutting plane. The tangent points of the spheres with the plane are the foci, and the tangent circles of the spheres with the cone define the directrices. This gives a purely geometric proof that the conic sections satisfy the focus-directrix definition.

Theorem 5 (Kepler's first law). Each planet moves in an elliptical orbit with the sun at one focus ^{[Kepler — Astronomia Nova]}. The eccentricity of Earth's orbit is approximately $0.017$ ; Mercury's is $0.206$ .

Theorem 6 (tangent line to a conic). The tangent to the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ at the point $(x_{0}, y_{0})$ on the curve is $\frac{x _{0} x}{a ^{2}} + \frac{y _{0} y}{b ^{2}} = 1$ . Analogous formulas hold for the parabola ( $y_{0} y = 2 p (x + x_{0})$ ) and the hyperbola ( $\frac{x _{0} x}{a ^{2}} - \frac{y _{0} y}{b ^{2}} = 1$ ).

Synthesis. The foundational reason the conic sections form a single family is the focus-directrix definition, which parametrises all three types by the continuous parameter $e$ . The central insight is that the discriminant $B^{2} - 4 A C$ of the general conic $A x^{2} + B x y + C y^{2} + \dots = 0$ is the algebraic shadow of the eccentricity, and the bridge is between the geometric slicing picture (angle of the cut) and the algebraic classification (sign of the discriminant). This is exactly the content that identifies the Apollonian theory of conic sections with the Cartesian theory of quadratic curves. Putting these together, the Dandelin-sphere construction proves the equivalence geometrically, the reflection properties follow from the focus-directrix definition via calculus or geometric optics, and Kepler's first law binds the whole theory to planetary motion. The pattern recurs in 00.11.01 where polar coordinates unify the conics as $r = l / (1 + e cos θ)$ , and the generalisation to projective geometry is the statement that all non-degenerate conics are equivalent under projective transformations — the discriminant vanishes precisely when the conic degenerates.

Full proof set [Master]

Proposition 1 (general conic discriminant classification). The conic $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ is an ellipse, parabola, or hyperbola according as $B^{2} - 4 A C < 0$ , $= 0$ , or $> 0$ .

Proof. The quadratic part $Q (x, y) = A x^{2} + B x y + C y^{2}$ has associated symmetric matrix $M = (A B /2 B /2 C)$ . Its determinant is $det M = A C - B^{2} /4 = - (B^{2} - 4 A C) /4$ . By Sylvester's law of inertia, the signature of $Q$ is determined by the signs of the eigenvalues of $M$ , and $det M$ is the product of the eigenvalues. If $Δ = B^{2} - 4 A C < 0$ then $det M > 0$ , so both eigenvalues have the same sign: $Q$ is definite and the conic is an ellipse. If $Δ = 0$ then $det M = 0$ , so one eigenvalue vanishes: $Q$ has rank $1$ and the conic is a parabola. If $Δ > 0$ then $det M < 0$ , so the eigenvalues have opposite signs: $Q$ is indefinite and the conic is a hyperbola. $□$

Proposition 2 (ellipse focal sum). The sum of distances from any point $P$ on the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ ( $a > b$ ) to the foci $(\pm c, 0)$ equals $2 a$ .

Proof. Given in Exercise 7 solution above. The key identity is $d_{1}^{2} = (a - c x / a)^{2}$ and $d_{2}^{2} = (a + c x / a)^{2}$ , from which $d_{1} + d_{2} = 2 a$ . $□$

Proposition 3 (Dandelin spheres for the parabola). When a plane cuts a cone parallel to a generator, a single Dandelin sphere fits between the cone and the plane, tangent to the plane at the focus and to the cone in a circle whose plane defines the directrix.

Proof. Consider a right circular cone with vertex $V$ and a cutting plane $Π$ parallel to a generator. There exists a unique sphere inscribed in the cone and tangent to $Π$ . The sphere touches $Π$ at a point $F$ (the focus) and touches the cone in a circle lying in a plane $Π^{'}$ . For any point $P$ on the intersection curve, the distance $d (P, F)$ equals the distance from $P$ to the tangent circle (both are tangent lengths from $P$ to the same sphere), and the distance from $P$ to the tangent circle equals the distance from $P$ to $Π^{'}$ measured along a generator of the cone, which equals $d (P, ℓ)$ where $ℓ$ is the line of intersection of $Π^{'}$ with $Π$ . Hence $d (P, F) = d (P, ℓ)$ and the curve is a parabola. $□$

Connections [Master]

Quadratic equations and the discriminant 00.03.02. The conic discriminant $B^{2} - 4 A C$ is the two-variable generalisation of the quadratic discriminant $b^{2} - 4 a c$ . The same trichotomy — positive, zero, negative — classifies the conic type just as it classifies the number of real roots of a quadratic. The bridge is that both discriminate by the signature of a quadratic form.
Cartesian coordinates and distance 00.09.01. The focus-directrix definition of conics builds directly on the Cartesian distance formula. Every conic equation is derived from a distance ratio $d (P, F) / d (P, ℓ)$ , and the standard forms are obtained by substituting the distance formula and simplifying. The coordinate-geometry framework from that unit is the algebraic engine behind every derivation in this one.
Unit-circle trigonometry 00.07.01. The parametrisation of the circle $x = cos θ$ , $y = sin θ$ from that unit generalises to parametrisations of ellipses ( $x = a cos t$ , $y = b sin t$ ) and hyperbolas ( $x = a sec t$ , $y = b tan t$ ), developed in 00.11.02. The polar form of conics in 00.11.01 uses the angular coordinate $θ$ from the unit circle to write all three types as $r = l / (1 + e cos θ)$ .

Historical & philosophical context [Master]

Apollonius of Perga ~200 BC wrote the eight-book treatise Conics ^{[Apollonius — Conics]}, systematically classifying the conic sections by the angle at which a plane cuts a cone and deriving hundreds of properties including the focus-directrix characterisation, the tangent constructions, and the conjugate-diameter theory. Apollonius introduced the terms ellipse, parabola, and hyperbola (from the Greek for "falling short", "comparison", and "exceeding"), reflecting the way the cutting plane compares to the side of the cone.

Johannes Kepler 1609 Astronomia Nova ^{[Kepler — Astronomia Nova]} showed that the orbit of Mars is an ellipse with the sun at one focus, overthrowing two millennia of circular-orbit astronomy and establishing the first of Kepler's three laws. Kepler's insight was that the conic sections, studied for their own sake by Apollonius, describe the actual paths of heavenly bodies. The reflection property of parabolas was exploited by Newton in his reflecting telescope design (1668), and the reflection property of hyperbolas underlies modern Cassegrain telescope optics. Girard Desargues 1639 and Blaise Pascal 1640 initiated projective geometry, in which the three conic types unify: a projective transformation maps any non-degenerate conic to any other, and the distinction between ellipse, parabola, and hyperbola is a matter of the position of the conic relative to the line at infinity.

Bibliography [Master]

@book{ApolloniusConics,
  author = {Apollonius of Perga},
  title = {Conics},
  note = {Books I--IV survive in Greek; Books V--VII in Arabic. English translation by R. C. Taliaferro, Green Lion Press 1998},
  year = {-200}
}

@book{Kepler1609,
  author = {Kepler, Johannes},
  title = {Astronomia Nova},
  publisher = {Voegelini},
  address = {Heidelberg},
  year = {1609}
}

@book{Lang1988,
  author = {Lang, Serge},
  title = {Basic Mathematics},
  publisher = {Springer},
  year = {1988}
}

@book{Apostol1967,
  author = {Apostol, Tom M.},
  title = {Calculus, Volume 1},
  publisher = {Wiley},
  edition = {2nd},
  year = {1967}
}