Conic Sections · study.

Slice a double cone with a plane and the edge of the cut is a conic section. The angle of the plane relative to the cone's axis decides the curve: a shallow cut gives an ellipse (a circle when horizontal), a cut parallel to the cone's side gives a parabola, and a steep cut through both nappes gives a hyperbola.

The three conics as plane sections of a double cone: a shallow cut gives an ellipse, a cut parallel to the slant gives a parabola, and a steep cut through both nappes gives a hyperbola.

The section definitions are geometric, stated in terms of distances to fixed points and lines. Each yields a clean equation once the curve is placed symmetrically on the axes.

Parabolas

Place the vertex at the origin with focus $F (0, p)$ and directrix $y = - p$ . A point $P (x, y)$ lies on the parabola when its distance to the focus equals its distance to the directrix.

A parabola with focus

F = (0, p)

and directrix

y = - p

: every point

P

is equidistant from the focus and the directrix, so

P F = P M

where

M

is the foot of the perpendicular to the directrix.

Equating $x^{2} + (y - p)^{2} = ∣ y + p ∣$ , squaring, and cancelling gives

x^{2} + (y - p)^{2} = (y + p)^{2} ⟹ x^{2} = 4 p y .

Ellipses

Put the foci at $(\pm c, 0)$ . The condition $∣ P F_{1} ∣ + ∣ P F_{2} ∣ = 2 a$ , squared twice to clear the radicals, reduces to $(a^{2} - c^{2}) x^{2} + a^{2} y^{2} = a^{2} (a^{2} - c^{2})$ . Since $c < a$ , set $b^{2} = a^{2} - c^{2} > 0$ and divide through.

An ellipse with foci

F_{1}, F_{2}

(\pm c, 0)

: for every point

P

the focal radii satisfy

∣ P F_{1} ∣ + ∣ P F_{2} ∣ = 2 a

; the semi-axes obey

b^{2} = a^{2} - c^{2}

If the foci coincide ( $c = 0$ ) then $a = b$ and the ellipse is a circle.

Ellipses and parabolas share a reflection property, a consequence of the equal angles a tangent makes with the focal radii. A ray leaving one focus of an ellipse reflects off the curve straight to the other focus; lithotripsy exploits this by placing a kidney stone at one focus and a shock-wave source at the other. A parabola is the limiting case with one focus sent to infinity: rays from the focus reflect into a parallel beam, which is why headlamp reflectors and satellite dishes have parabolic cross-sections.

Reflection property of the ellipse: a ray from focus

F_{1}

meets the curve at

P

and reflects to

F_{2}

, since the tangent at

P

makes equal angles with the two focal radii.

Hyperbolas

The derivation mirrors the ellipse, but now $c > a$ , so $b^{2} = c^{2} - a^{2} > 0$ . The curve has two branches and no $y$ -intercept, and each branch approaches a pair of straight asymptotes.

A hyperbola

x^{2} / a^{2} - y^{2} / b^{2} = 1

with foci at

(\pm c, 0)

c^{2} = a^{2} + b^{2}

: the two branches open along the

x

-axis and hug the asymptotes

y = \pm (b / a) x

Shifted conics. Replacing $x$ by $x - h$ and $y$ by $y - k$ moves the center to $(h, k)$ ; completing the square recovers the standard form from a general second-degree equation.

Conic	Focal condition	Standard equation	Key relation
Parabola	$∣ P F ∣ = ∣ P ℓ ∣$	$x^{2} = 4 p y$	vertex midway to directrix
Ellipse	$∣ P F_{1} ∣ + ∣ P F_{2} ∣ = 2 a$	$\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$	$b^{2} = a^{2} - c^{2}$
Hyperbola	$∣ P F_{1} ∣ - ∣ P F_{2} ∣ = 2 a$	$\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$	$b^{2} = c^{2} - a^{2}$

Eccentricity

The focus-and-directrix definition of the parabola generalizes to all conics by allowing the distance ratio to differ from $1$ .

The focus-directrix ratio

∣ P F ∣/∣ P ℓ ∣ = e

defines every conic at once; the value of

e

selects the type, with

e < 1

an ellipse,

e = 1

a parabola, and

e > 1

a hyperbola.

For an ellipse $e = c / a < 1$ , and for a hyperbola $e = c / a > 1$ . Eccentricity is a shape number, independent of size: $e$ near $0$ is nearly circular, $e$ near $1$ is elongated, and $e$ growing past $1$ opens the branches of a hyperbola wider.

Eccentricity	Conic
$e = 0$	circle
$0 < e < 1$	ellipse
$e = 1$	parabola
$e > 1$	hyperbola

The polar equation of a conic

Placing the focus at the pole turns the ratio definition into one compact polar equation. With directrix $x = d$ to the right of the focus, $∣ P F ∣ = r$ and $∣ P ℓ ∣ = d - r cos θ$ , so $r = e (d - r cos θ)$ . Solving for $r$ :¹

r = \frac{e d}{1 + e cos θ} .

The four sign-and-function combinations correspond to the four directrix placements relative to the focus:

$r = \frac{e d}{1 + e cos θ}$ : directrix vertical, to the right of the focus.
$r = \frac{e d}{1 - e cos θ}$ : directrix vertical, to the left.
$r = \frac{e d}{1 + e sin θ}$ : directrix horizontal, above the focus.
$r = \frac{e d}{1 - e sin θ}$ : directrix horizontal, below.

Orbits

Kepler's first law places each planet on an ellipse with the sun at one focus. Writing the polar equation in terms of the semimajor axis $a$ and eccentricity $e$ (using $e d = a (1 - e^{2})$ ) gives the standard orbital form.

r = \frac{a ( 1 - e ^{2} )}{1 + e cos θ} .

The extreme distances occur at the vertices. At $θ = 0$ the planet is at perihelion (closest), and at $θ = π$ at aphelion (farthest):

r_{min} = a (1 - e), r_{max} = a (1 + e) .

For Earth, $e \approx 0.017$ and $2 a \approx 2.99 \times 1 0^{8}$ km, so $a \approx 1.495 \times 1 0^{8}$ km. The orbit is $r \approx \frac{1.49 \times 1 0 ^{8}}{1 + 0.017 cos θ}$ km, with perihelion about $1.47 \times 1 0^{8}$ km and aphelion about $1.52 \times 1 0^{8}$ km. A small eccentricity keeps the orbit close to a circle, and the same equation, with $e$ pushed past $1$ , describes the unbound hyperbolic path of a comet that swings once past the sun and never returns.

Stewart, §10.6 — the polar conic equation follows from the focus-directrix ratio $∣ P F ∣ = e ∣ P ℓ ∣$ with the focus at the pole and the directrix perpendicular to the polar axis. ↩