00.10.02 · precalc / conics

Conic sections — parabola, ellipse, hyperbola

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Anchor (Master): Apollonius c.200 BCE On Conics; Coolidge 1945 A History of the Conic Sections and Geometrical Topics — the symptom relations and the projective unification

Intuition Beginner

Picture a double cone — two ice-cream cones glued tip to tip. Now slice it with a flat plane. Four things can happen depending on the angle of the cut. Cut straight across and you get a circle. Tilt the blade a little and the circle stretches into an oval, the ellipse. Tilt until the blade runs parallel to the side of the cone and the curve never closes back up — it is a parabola. Tilt even steeper so the blade cuts both cones and you get two outward-facing curves, the hyperbola.

Here is the deeper surprise. All four curves follow the very same rule. Pick a fixed point called the focus and a fixed line called the directrix. For every point on the curve, divide its distance to the focus by its distance to the directrix. That ratio is a single constant number, the eccentricity $e$ . A ratio of $1$ gives the parabola; below $1$ gives the ellipse (and the circle is the case $e = 0$ ); above $1$ gives the hyperbola.

One rule, three curves, and the number $e$ alone decides which one you draw. This unit builds the parabola, the ellipse, and the hyperbola in full detail by starting from that one ratio rule and turning it into an equation you can graph.

Visual Beginner

The table collects the four slices, the curve each one produces, and the eccentricity that labels it. Reading down the table, the eccentricity grows from $0$ (a perfect circle) upward through the ellipses, hits $1$ at the parabola, and keeps climbing through the hyperbolas.

Cut angle	Curve	Eccentricity $e$
Flat, across the axis	Circle	$0$
Tilted, shallow	Ellipse	$0 < e < 1$
Parallel to the side	Parabola	$1$
Steeper than the side	Hyperbola	$e > 1$

The same four curves, drawn from the focus-directrix rule, land in the same order. As $e$ slides from $0$ upward the shape stretches, opens, and finally splits into two branches.

Worked example Beginner

Example 1. Identify and sketch $y^{2} = 8 x$ .

Step 1. The equation has a $y^{2}$ term and a single $x$ term, but no $x^{2}$ . That signature is a parabola.

Step 2. Match it against the standard parabola form $y^{2} = 4 p x$ . The coefficient of $x$ gives $4 p = 8$ , so $p = 2$ .

Step 3. Because $x$ is positive on the right side and the squared variable is $y$ , the parabola opens to the right. Its vertex sits at the origin and its focus is the point $(2, 0)$ . The directrix is the vertical line $x = - 2$ .

Example 2. Identify and sketch $x^{2} /25 + y^{2} /9 = 1$ .

Step 1. Both $x^{2}$ and $y^{2}$ appear and they share the same sign. That signature is an ellipse.

Step 2. Read off the two semi-axes straight from the denominators: $a = 5$ and $b = 3$ .

Step 3. The focal distance obeys $c^{2} = a^{2} - b^{2} = 25 - 9 = 16$ , so $c = 4$ . The foci are the two points $(\pm 4, 0)$ on the long axis. The ellipse is a stretched circle, $10$ wide and $6$ tall, centred at the origin.

Both answers fall out the same way: read the signs and the squared terms to name the curve, then read off the parameters to place its key points.

Check your understanding Beginner

Formal definition Intermediate+

A conic section is the locus of a point $P$ that keeps a fixed ratio between its distance to a fixed point $F$ (the focus) and its distance to a fixed line $ℓ$ (the directrix), with $F \in / ℓ$ . The ratio is the eccentricity $e \geq 0$ :

\frac{d ( P , F )}{d ( P , ℓ )} = e .

The value of $e$ alone names the curve. When $e = 1$ the locus is a parabola; when $0 \leq e < 1$ it is an **ellipse** (and $e = 0$ collapses it to a **circle**); when $e > 1$ it is a hyperbola. Each curve also admits a second, two-foci characterisation that is often easier to use.

Parabola. Place the focus at $F = (p, 0)$ and the directrix at $x = - p$ , with $p \neq = 0$ . The locus $d (P, F) = d (P, ℓ)$ gives the standard form

y^{2} = 4 p x,

with vertex at the origin and the line through $F$ perpendicular to $ℓ$ (here the $x$ -axis) as the axis of symmetry. Translating the vertex to $(h, k)$ gives the two shifted forms

y = a (x - h)^{2} + k (vertical axis), x = a (y - k)^{2} + h (horizontal axis),

where $a = 1/ (4 p)$ . The sign of $a$ fixes the opening direction: $a > 0$ opens the vertical form upward and the horizontal form to the right.

Ellipse. The ellipse with foci $F_{1}, F_{2}$ is the locus of points $P$ with

d (P, F_{1}) + d (P, F_{2}) = 2 a,

a constant larger than the distance between the foci. With foci at $(\pm c, 0)$ and $c^{2} = a^{2} - b^{2}$ , its standard form is

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b > 0,

with semi-major axis $a$ , semi-minor axis $b$ , and eccentricity $e = c / a \in [0, 1)$ . The two descriptions are equivalent: the focus-directrix ellipse and the constant-sum ellipse are the same curve ^{[Stewart — Precalculus: Mathematics for Calculus]}.

Hyperbola. The hyperbola with foci $F_{1}, F_{2}$ is the locus of points $P$ for which the absolute difference is constant:

∣ d (P, F_{1}) - d (P, F_{2}) ∣ = 2 a .

With foci at $(\pm c, 0)$ and $c^{2} = a^{2} + b^{2}$ , the standard form is

\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1,

with transverse semi-axis $a$ , conjugate semi-axis $b$ , eccentricity $e = c / a > 1$ , and asymptotes $y = \pm (b / a) x$ . The curve has two branches opening left and right, each hugging its asymptotes at infinity.

Counterexamples to common slips

The standard form $y^{2} = 4 p x$ opens to the right only when $p > 0$ ; with $p < 0$ it opens to the left. The squared variable, not the sign alone, fixes which axis the parabola spans, and the sign of $p$ (or $a$ ) fixes the direction along that axis.
In the ellipse $x^{2} / a^{2} + y^{2} / b^{2} = 1$ the larger denominator labels the major axis. If $b > a$ the major axis is vertical and the foci are at $(0, \pm c)$ with $c^{2} = b^{2} - a^{2}$ , not $(\pm c, 0)$ .
The hyperbola $x^{2} / a^{2} - y^{2} / b^{2} = 1$ opens left and right because the $x^{2}$ term is positive. Swapping the signs to $y^{2} / b^{2} - x^{2} / a^{2} = 1$ opens it up and down, and the foci move to the $y$ -axis. The two equations are conjugate hyperbolas, not the same curve.

Key derivation Intermediate+

This section derives every standard form above from a single starting point: a distance condition. The first derivation handles the parabola; the next two build the ellipse and hyperbola from their two-foci definitions, which themselves follow from the focus-directrix rule.

Derivation 1 (parabola, vertex at the origin). Take the focus $F = (p, 0)$ and the directrix $ℓ : x = - p$ . For $P = (x, y)$ the definition $d (P, F) = d (P, ℓ)$ reads

(x - p)^{2} + y^{2} = ∣ x + p ∣ .

Square both sides and expand:

(x - p)^{2} + y^{2} = (x + p)^{2} ⟹ x^{2} - 2 p x + p^{2} + y^{2} = x^{2} + 2 p x + p^{2} .

Cancel $x^{2}$ and $p^{2}$ and collect: $y^{2} = 4 p x$ . Translating the vertex to $(h, k)$ by the substitution $x \mapsto x - h$ , $y \mapsto y - k$ gives $(y - k)^{2} = 4 p (x - h)$ , i.e. $y = \frac{1}{4 p} (x - h)^{2} + k$ . Setting $a = 1/ (4 p)$ recovers the vertical-axis form $y = a (x - h)^{2} + k$ . Swapping the roles of $x$ and $y$ (focus on the $y$ -axis, horizontal directrix) gives the horizontal-axis form $x = a (y - k)^{2} + h$ with the same relation $a = 1/ (4 p)$ .

Derivation 2 (ellipse, constant sum $2 a$ ). Take foci $F_{1} = (c, 0)$ , $F_{2} = (- c, 0)$ with $0 \leq c < a$ and impose $d (P, F_{1}) + d (P, F_{2}) = 2 a$ . Write $d_{1} = d (P, F_{1})$ , $d_{2} = d (P, F_{2})$ . The defining relation and its rearrangement give

d_{1} = 2 a - d_{2} ⟹ d_{1}^{2} = 4 a^{2} - 4 a d_{2} + d_{2}^{2} .

Substitute the squared distance formulas $d_{1}^{2} = (x - c)^{2} + y^{2}$ and $d_{2}^{2} = (x + c)^{2} + y^{2}$ and simplify:

(x - c)^{2} + y^{2} = 4 a^{2} - 4 a d_{2} + (x + c)^{2} + y^{2} .

Cancel $y^{2}$ , expand the squares, and collect to isolate the surviving square root:

- 4 c x = 4 a^{2} - 4 a d_{2} ⟹ d_{2} = a + \frac{c x}{a} .

Square once more, $d_{2}^{2} = (a + c x / a)^{2}$ , and equate to $(x + c)^{2} + y^{2}$ :

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

Cancel $2 c x$ , collect the $x^{2}$ terms, and use $c^{2} = a^{2} - b^{2}$ :

\frac{a ^{2} - c ^{2}}{a ^{2}} x^{2} + y^{2} = a^{2} - c^{2} ⟹ \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1,

where $b^{2} = a^{2} - c^{2}$ . The relation $e = c / a$ then reads $e^{2} = 1 - b^{2} / a^{2}$ , confirming $0 \leq e < 1$ . Reversing the algebra shows that every point on this curve satisfies the constant-sum condition, so the two characterisations coincide.

Derivation 3 (hyperbola, constant difference $2 a$ ). With the same foci $F_{1} = (c, 0)$ , $F_{2} = (- c, 0)$ but now $c > a > 0$ , impose $∣ d_{1} - d_{2} ∣ = 2 a$ . On the right branch $d_{2} - d_{1} = 2 a$ . The same squaring dance as above, with $- 2 a$ in place of $2 a$ , yields

d_{2} = - a + \frac{c x}{a} (right branch),

and after squaring and collecting with $c^{2} = a^{2} + b^{2}$ ,

\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1.

The asymptotes fall out by division: rewrite as $y^{2} = b^{2} (x^{2} / a^{2} - 1)$ , so $y = \pm (b / a) x^{2} - a^{2}$ . As $∣ x ∣ \to \infty$ the term $- a^{2}$ under the root becomes negligible and $y \to \pm (b / a) ∣ x ∣$ , giving the asymptotes $y = \pm (b / a) x$ .

Bridge. The focus-directrix derivation builds toward 00.11.01, where the single polar equation $r = ℓ / (1 + e cos θ)$ reproduces all three standard forms at once, and appears again in 00.11.02 as the rational parametrisation of each conic. The foundational reason one distance-ratio produces three distinct curves is that the algebraic sign of $1 - e^{2}$ selects which branch of the quadratic the locus occupies — this is exactly the content of the three derivations above, where $b^{2} = a^{2} - c^{2}$ (ellipse) versus $b^{2} = c^{2} - a^{2}$ (hyperbola) is the central insight, and the parabola sits between them as the case $c^{2} = a^{2}$ that cancels the constant term. Putting these together, the bridge is between the geometric definition (a ratio or sum of distances) and the algebraic equation (a degree-two curve in $x$ and $y$ ), and the pattern generalises to the discriminant test of 00.10.01, which classifies any quadratic by a single number $Δ = B^{2} - 4 A C$ .

Exercises Intermediate+

Exercise 6 (medium, numeric).

A whispering gallery is a room whose horizontal cross-section is an ellipse. The room is $100$ feet long and $40$ feet wide. Two people standing at the foci can hear each other with perfect clarity while listeners elsewhere cannot. Where should the two people stand? Give the distance of each from the centre of the room, to the nearest foot.

Hint

The length is the major axis $2 a$ and the width is the minor axis $2 b$ . Use $c^{2} = a^{2} - b^{2}$ .

Answer

Each person stands about $46$ feet from the centre, along the long axis.

The semi-axes are $a = 50$ and $b = 20$ . The focal distance is $c = 5 0^{2} - 2 0^{2} = 2500 - 400 = 2100 \approx 45.83$ feet, which rounds to $46$ . So the two people stand at $(\pm 46, 0)$ relative to the centre. The reflection property of the ellipse gathers every sound ray leaving one focus onto the other, which is why the whisper carries.

Rubric: full credit for computing $c$ and rounding sensibly.

Exercise 7 (hard, symbolic).

Derive the standard form of the hyperbola from the difference-of-distances definition. Take foci $F_{1} = (c, 0)$ , $F_{2} = (- c, 0)$ with $c > a > 0$ , impose $∣ d_{1} - d_{2} ∣ = 2 a$ on the right branch, and show the locus is $x^{2} / a^{2} - y^{2} / b^{2} = 1$ with $b^{2} = c^{2} - a^{2}$ .

Hint

On the right branch $d_{2} - d_{1} = 2 a$ . Isolate one square root, square, simplify, and use $b^{2} = c^{2} - a^{2}$ at the end.

Answer

On the right branch $d_{2} - d_{1} = 2 a$ , so $d_{1} = d_{2} - 2 a$ and squaring gives $d_{1}^{2} = d_{2}^{2} - 4 a d_{2} + 4 a^{2}$ . Substitute the squared distance formulas:

(x - c)^{2} + y^{2} = (x + c)^{2} + y^{2} - 4 a d_{2} + 4 a^{2} .

Cancel $y^{2}$ , expand, and collect: $- 4 c x = - 4 a d_{2} + 4 a^{2}$ , so $d_{2} = a + c x / a$ . Squaring again, $d_{2}^{2} = (a + c x / a)^{2} = a^{2} + 2 c x + c^{2} x^{2} / a^{2}$ . Equate to $(x + c)^{2} + y^{2} = x^{2} + 2 c x + c^{2} + y^{2}$ and cancel $2 c x$ :

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

Multiply through by $a^{2}$ and collect the $x^{2}$ term: $(c^{2} - a^{2}) x^{2} - a^{2} y^{2} = a^{2} (c^{2} - a^{2})$ . Setting $b^{2} = c^{2} - a^{2} > 0$ and dividing by $a^{2} b^{2}$ yields $x^{2} / a^{2} - y^{2} / b^{2} = 1$ .

Rubric: full credit for the two squarings, the simplification, and the identification $b^{2} = c^{2} - a^{2}$ .

Exercise 8 (hard, symbolic).

Show that the hyperbola $x^{2} / a^{2} - y^{2} / b^{2} = 1$ has asymptotes $y = \pm (b / a) x$ . Specifically, prove that for each choice of sign the vertical distance from a point on the curve to the line $y = (b / a) x$ tends to $0$ as $x \to + \infty$ .

Hint

Solve the standard form for $y$ on the upper branch, subtract the line, and bound the result after rationalising.

Answer

On the upper branch $y = + (b / a) x^{2} - a^{2}$ . The vertical gap to the line $y = (b / a) x$ is

δ (x) = \frac{b}{a} (x - x^{2} - a^{2}) = \frac{b}{a} \cdot \frac{a ^{2}}{x + x ^{2} - a ^{2}},

where the second equality rationalises by multiplying by $(x + x^{2} - a^{2}) / (x + x^{2} - a^{2})$ . As $x \to + \infty$ the denominator grows without bound, so $δ (x) \to 0$ . The lower branch $y = - (b / a) x^{2} - a^{2}$ is the mirror image and approaches $y = - (b / a) x$ by the same argument. Hence both asymptotes are $y = \pm (b / a) x$ .

Rubric: full credit for solving for $y$ , rationalising the gap, and arguing the limit.

Advanced results Master

Theorem 1 (focal-sum and focal-difference characterisations). The ellipse is the unique locus with $d (P, F_{1}) + d (P, F_{2}) = 2 a > d (F_{1}, F_{2})$ ; the hyperbola is the unique locus with $∣ d (P, F_{1}) - d (P, F_{2})∣ = 2 a < d (F_{1}, F_{2})$ . These two-foci definitions are equivalent to the single-focus-directrix definition with $e = c / a$ : every focus-directrix ellipse has a second focus (the reflection of the first through the centre), and every focus-directrix hyperbola likewise.

Theorem 2 (reflection properties). A ray leaving one focus of an ellipse reflects off the curve to pass through the other focus; a ray leaving the focus of a parabola reflects parallel to the axis; and a ray aimed at one focus of a hyperbola reflects as though coming from the other. All three follow from the equal-angles law of reflection combined with the tangent line to the curve, and they underlie telescope, antenna, and lithotripsy design.

Theorem 3 (tangent lines). The tangent to the ellipse $x^{2} / a^{2} + y^{2} / b^{2} = 1$ at $(x_{0}, y_{0})$ is $x_{0} x / a^{2} + y_{0} y / b^{2} = 1$ . The tangent to the parabola $y^{2} = 4 p x$ at $(x_{0}, y_{0})$ is $y_{0} y = 2 p (x + x_{0})$ . The tangent to the hyperbola $x^{2} / a^{2} - y^{2} / b^{2} = 1$ at $(x_{0}, y_{0})$ is $x_{0} x / a^{2} - y_{0} y / b^{2} = 1$ . Each formula is the polar line of the point of tangency and doubles as the equation of the chord of contact from an external point.

Theorem 4 (polar unification). With the focus at the origin and the directrix at $x = ℓ / e$ , every conic of eccentricity $e$ satisfies

r = \frac{ℓ}{1 + e cos θ},

where $ℓ$ is the semi-latus rectum. Substituting $e < 1$ , $e = 1$ , $e > 1$ recovers the ellipse, parabola, and hyperbola standard forms, so the polar equation packages the entire eccentricity classification into a single expression ^{[Stewart — Precalculus: Mathematics for Calculus]}.

Theorem 5 (Dandelin spheres). Inside each nappe of the cone one can inscribe a sphere tangent to both the cone and the cutting plane. The tangent point of such a sphere with the cutting plane is a focus of the conic, and the circle of tangency with the cone lies in a plane whose intersection with the cutting plane is a directrix. The Dandelin construction gives a purely geometric proof that the cone-slicing definition and the focus-directrix definition describe the same curves.

Synthesis. The foundational reason the three conics form one family is the unified focus-directrix definition, parametrised by the single continuous value $e$ . The central insight is that each conic's signature property — the parabola's equal distances, the ellipse's constant sum $2 a$ , the hyperbola's constant difference $2 a$ — is exactly the geometric shadow of one algebraic identity, and this generalises immediately to the polar form $r = ℓ / (1 + e cos θ)$ that reproduces all three standard forms from one equation. The reflection properties and tangent-line formulas follow from the same focus-directrix seed via the equal-angles law, and putting these together, the bridge is between Apollonius's synthetic slicing picture and Descartes's algebraic picture of degree-two curves. The pattern generalises once more in projective geometry, where the distinction between ellipse, parabola, and hyperbola dissolves: a projective transformation carries any non-degenerate conic to any other, and the three types record only how the curve meets the line at infinity.

Full proof set Master

Proposition 1 (ellipse focal sum). The sum of the distances from any point $P$ on the ellipse $x^{2} / a^{2} + y^{2} / b^{2} = 1$ ( $a > b$ ) to the foci $F_{1} = (c, 0)$ , $F_{2} = (- c, 0)$ , where $c^{2} = a^{2} - b^{2}$ , is the constant $2 a$ .

Proof. Take $P = (x, y)$ on the ellipse, so $y^{2} = b^{2} (1 - x^{2} / a^{2})$ and $∣ x ∣ \leq a$ . Define

d_{1}^{2} = (x - c)^{2} + y^{2}, d_{2}^{2} = (x + c)^{2} + y^{2} .

Substitute $y^{2} = b^{2} - (b^{2} / a^{2}) x^{2}$ and use $b^{2} = a^{2} - c^{2}$ :

d_{1}^{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 $1 - b^{2} / a^{2} = c^{2} / a^{2}$ and $c^{2} + b^{2} = a^{2}$ , this becomes

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

Because $c < a$ and $∣ x ∣ \leq a$ , the quantity $a - c x / a$ is non-negative, so $d_{1} = a - c x / a$ . A symmetric computation gives $d_{2} = a + c x / a$ . Adding,

d_{1} + d_{2} = (a - \frac{c x}{a}) + (a + \frac{c x}{a}) = 2 a . □

Proposition 2 (hyperbola asymptotes). The hyperbola $x^{2} / a^{2} - y^{2} / b^{2} = 1$ has asymptotes $y = \pm (b / a) x$ .

Proof. Solve the standard form for $y$ on the upper branch: $y = (b / a) x^{2} - a^{2}$ , defined for $∣ x ∣ \geq a$ . Compare with the line $L (x) = (b / a) x$ . Their difference, rationalised, is

L (x) - y = \frac{b}{a} (x - x^{2} - a^{2}) = \frac{b}{a} \cdot \frac{a ^{2}}{x + x ^{2} - a ^{2}} .

As $x \to + \infty$ the denominator $x + x^{2} - a^{2}$ grows without bound, so the difference tends to $0$ . The upper branch therefore approaches the line $y = (b / a) x$ . The lower branch $y = - (b / a) x^{2} - a^{2}$ is the reflection through the $x$ -axis and approaches $y = - (b / a) x$ by the same argument. Both asymptotes are thus $y = \pm (b / a) x$ . $□$

Proposition 3 (horizontal parabola from focus and directrix). The locus of points equidistant from the focus $F = (h + p, k)$ and the directrix $ℓ : x = h - p$ (a vertical line, perpendicular to the horizontal axis of symmetry) is the parabola $x = \frac{1}{4 p} (y - k)^{2} + h$ , i.e. the horizontal-axis standard form $x = a (y - k)^{2} + h$ with $a = 1/ (4 p)$ .

Proof. For $P = (x, y)$ the defining condition $d (P, F) = d (P, ℓ)$ reads

(x - h - p)^{2} + (y - k)^{2} = ∣ x - h + p ∣ .

Square both sides and expand each quadratic in $x - h$ :

(x - h)^{2} - 2 p (x - h) + p^{2} + (y - k)^{2} = (x - h)^{2} + 2 p (x - h) + p^{2} .

Cancel $(x - h)^{2}$ and $p^{2}$ from both sides:

- 2 p (x - h) + (y - k)^{2} = 2 p (x - h) .

Collect the $x - h$ terms: $(y - k)^{2} = 4 p (x - h)$ . Solve for $x$ :

x = \frac{1}{4 p} (y - k)^{2} + h .

Setting $a = 1/ (4 p)$ recovers the horizontal-axis standard form $x = a (y - k)^{2} + h$ . The sign of $p$ (equivalently of $a$ ) fixes the opening direction: $p > 0$ opens the parabola to the right, $p < 0$ to the left. $□$

Connections Master

The conic classification and discriminant 00.10.01. This unit is the deep companion to 00.10.01: the sibling unit classifies any quadratic $A x^{2} + B x y + C y^{2} + \dots = 0$ by the single number $Δ = B^{2} - 4 A C$ , while the present unit derives each conic's standard form by hand from a distance condition. Read together, the two units show that the geometric focus-directrix picture and the algebraic discriminant picture describe one and the same family of curves.
Cartesian coordinates and the distance formula 00.09.01. Every derivation in this unit runs on the distance formula $(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$ from 00.09.01. The focus-directrix condition $d (P, F) / d (P, ℓ) = e$ becomes an equation only once Cartesian coordinates translate "distance" into square roots of sums of squares, and completing the square is exactly the coordinate-algebra move that recentres the curve at its vertex or centre.
Polar coordinates and the unified conic 00.11.01. The polar equation $r = ℓ / (1 + e cos θ)$ derived in 00.11.01 packages the three standard forms of this unit into a single expression indexed by $e$ . The focus placed at the pole in that unit is the same focus that appears in the focus-directrix definition here, so the polar form is the natural coordinate change that turns the distance-ratio rule directly into a graph.
Parametric and rational curves 00.11.02. Each standard form derived here admits a rational parametrisation developed in 00.11.02 (for instance the parabola $y^{2} = 4 p x$ parametrises as $(p t^{2}, 2 pt)$ , and the ellipse $x^{2} / a^{2} + y^{2} / b^{2} = 1$ as $(a cos t, b sin t)$ ). Those parametrisations convert the implicit equations of this unit into the rational maps that underpin projective geometry and Bezier-style curve design.

Historical & philosophical context Master

Menaechmus of Proconnesus, around 350 BCE, is credited with the discovery of the conic sections while working on the problem of doubling the cube ^{[Coolidge — A History of the Conic Sections and Geometrical Topics]}. He obtained the three curves by slicing a cone at three different fixed angles, and the resulting "symptom" relations — the precursor of our standard forms — gave the first geometric solution to the Delian problem of constructing $32$ . Menaechmus's account survives only through later commentators; the direct textual tradition begins with Apollonius.

Apollonius of Perga, around 200 BCE, codified the theory in the eight-book treatise On Conic Sections ^{[Apollonius of Perga — On Conic Sections]}. Apollonius proved that all three curves can be cut from a single cone by varying the angle of the plane, derived the focus-directrix characterisation, developed the theory of conjugate diameters, and named the curves ellipse, parabola, and hyperbola from the Greek verbs for "falling short", "application", and "exceeding" — describing how the cutting plane compares to a generator of the cone. The Apollonian names survive unchanged in the modern standard forms $x^{2} / a^{2} + y^{2} / b^{2} = 1$ , $y^{2} = 4 p x$ , and $x^{2} / a^{2} - y^{2} / b^{2} = 1$ .

Johannes Kepler, in the Astronomia Nova of 1609, established that the orbit of Mars is an ellipse with the sun at one focus, displacing two millennia of circular-orbit astronomy and turning the conic sections from a pure geometry into the language of celestial mechanics. Kepler's first law binds the focus-directrix theory of this unit to the physical world: the eccentricity $e = c / a$ that labels a conic on paper is the same $e$ that measures how far a planet's orbit departs from a circle, with Earth at $e \approx 0.017$ and Mercury at $e \approx 0.206$ . René Descartes's 1637 coordinate reformulation recast the Apollonian curves as the degree-two equations $A x^{2} + B x y + C y^{2} + \dots = 0$ studied in 00.10.01, and the nineteenth-century projective geometers — Poncelet, Steiner, Chasles — finally unified the three types by showing that any non-degenerate conic can be carried to any other by a projective transformation.

Bibliography Master

@book{StewartPrecalculus2012,
  author    = {Stewart, James and Redlin, Lothar and Watson, Saleem},
  title     = {Precalculus: Mathematics for Calculus},
  edition   = {7th},
  publisher = {Cengage Learning},
  year      = {2012}
}

@book{ApolloniusConicsTal,
  author    = {Apollonius of Perga},
  title     = {On Conic Sections},
  note      = {Books I--IV in Greek, V--VII from Arabic. English translation by R. Catesby Taliaferro, Green Lion Press, 1998},
  year      = {-200}
}

@book{CoolidgeConics1945,
  author    = {Coolidge, Julian Lowell},
  title     = {A History of the Conic Sections and Geometrical Topics},
  publisher = {Dover Publications},
  year      = {1945}
}

@book{KeplerAstronomiaNova1609,
  author    = {Kepler, Johannes},
  title     = {Astronomia Nova},
  publisher = {Voegelini},
  address   = {Heidelberg},
  year      = {1609}
}

@book{DescartesGeometry1637,
  author    = {Descartes, Ren{\'e}},
  title     = {La G{\'e}om{\'e}trie},
  publisher = {Jan Maire},
  address   = {Leiden},
  year      = {1637}
}

Prerequisites

00.10.01
00.09.01

Tier anchors

beginner: Stewart 2012 Precalculus: Mathematics for Calculus (Cengage) Ch. 9 — the four slices of a cone, with the focus/directrix picture drawn for each curve
intermediate: Stewart 2012 Precalculus (Cengage) §9.1–9.5 — standard forms derived from the focus-directrix condition and from the sum/difference of distances
master: Apollonius c.200 BCE On Conics; Coolidge 1945 A History of the Conic Sections and Geometrical Topics — the symptom relations and the projective unification

References

Stewart — Precalculus: Mathematics for Calculus · Cengage (2012), 7th ed. Chapter 9, Conic Sections. Stewart builds each conic first as a plane cut of a cone, then derives the standard Cartesian forms from the focus-directrix condition. Section 9.1 covers the parabola with both standard forms $y^{2} = 4 p x$ and $y = a (x - h)^{2} + k$ ; section 9.2 the ellipse with $x^{2} / a^{2} + y^{2} / b^{2} = 1$ , $c^{2} = a^{2} - b^{2}$ , eccentricity $e = c / a$ ; section 9.3 the hyperbola with $x^{2} / a^{2} - y^{2} / b^{2} = 1$ , $c^{2} = a^{2} + b^{2}$ , asymptotes $y = \pm (b / a) x$ ; section 9.4 the unified focus-directrix definition with eccentricity; section 9.5 shifted conics by completing the square.
Apollonius of Perga — On Conic Sections · c.200 BCE, Books I through IV in Greek, with V through VII surviving in Arabic. English translation by R. Catesby Taliaferro, Green Lion Press 1998. Book I defines the three conic sections as plane cuts of a cone and proves the fundamental symptom — the relation we now write $y^{2} = 2 p x$ for the parabola and $y^{2} = 2 p x \pm (p / a) x^{2}$ for the hyperbola and ellipse. Apollonius coins the names ellipse, parabola, hyperbola from the Greek for "falling short", "application", and "exceeding", reflecting how the cutting plane compares to the side of the cone.
Coolidge — A History of the Conic Sections and Geometrical Topics · Dover (1945). Coolidge traces the Apollonian theory through Kepler 1609 elliptical orbits, Descartes coordinate reformulation of the conics as degree-two curves $A x^{2} + B x y + C y^{2} + \dots = 0$ , and into the projective unification under which all non-degenerate conics are equivalent over the complex numbers.

Estimated time

beginner: 20m
intermediate: 50m
master: 95m