The Four Conic Sections | Moosa Academy

The Four Conic Sections — Moosa Academy

Slice a cone with a flat plane and the edge of the cut traces a curve. Tilt the plane and the curve changes shape. Every possible result belongs to one of four families — the circle, ellipse, parabola, and hyperbola — and a single number, the eccentricity, tells you which one you have.

Concept Where the four shapes come from

Cut perpendicular to the axis — a circle.

Tilt it slightly — an ellipse.

Tilt until parallel to the cone's side — a parabola.

Tilt steeper still — a hyperbola.

The steeper the cutting plane, the larger the eccentricity of the resulting curve.

Concept The circle

(x - h)^2 + (y - k)^2 = r^2

A circle is the set of all points at a fixed distance $r$ from a fixed centre $(h,\ k)$ .

Circumference

= 2\pi r

, area

= \pi r^2

Eccentricity

e = 0

— the most perfectly round case.

Concept The ellipse

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1

An ellipse is the set of points for which the sum of the distances to two fixed points (the foci) is constant.

a

and

b

are the semi-axes; the focal distance satisfies

c^2 = a^2 - b^2

Eccentricity

e = c/a

, always between 0 and 1.

When

a = b

the two foci merge at the centre and the ellipse becomes a circle.

Concept The parabola

(x - h)^2 = 4p(y - k)

A parabola is the set of points equally distant from a fixed point (the focus) and a fixed line (the directrix).

(h,\ k)

is the vertex and

p

is the distance from the vertex to the focus.

Swapping the roles of

x

and

y

gives

(y-k)^2 = 4p(x-h)

, a parabola opening sideways.

Eccentricity

e = 1

— exactly one focus, no second branch.

Concept The hyperbola

\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1

A hyperbola is the set of points for which the absolute difference of the distances to two fixed foci is constant. The minus sign is the only thing separating this equation from the ellipse — and it changes everything.

Here

c^2 = a^2 + b^2

, so

c

is larger than

a

and the eccentricity

e = c/a

always exceeds 1.

The curve has two separate branches and two asymptotes,

y - k = \pm\dfrac{b}{a}(x - h)

Note The four compared

Shape	Eccentricity	Defining property	Foci
Circle	e = 0	Fixed distance from a centre	One centre
Ellipse	0 < e < 1	Sum of distances is constant	Two
Parabola	e = 1	Equal distance to focus and directrix	One
Hyperbola	e > 1	Difference of distances is constant	Two, curve in two branches

Example Identifying a conic from its equation

Classify $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$ , and find its eccentricity.

Both squared terms are positive and added, with different denominators — an ellipse centred at the origin.

a^2 = 25 \Rightarrow a = 5

, and

b^2 = 9 \Rightarrow b = 3

c^2 = a^2 - b^2 = 25 - 9 = 16

, so

c = 4

⟹ ellipse, e = 4/5 = 0.8

Had the sign between the terms been a minus, the very same numbers would have described a hyperbola instead, with $c^2 = 25 + 9 = 34$ .

Note Where these curves appear

Circle — wheels, gears, pipes, clock faces.

Ellipse — planetary and satellite orbits, elliptical domes and arches.

Parabola — projectile paths, satellite dishes, headlight reflectors.

Hyperbola — cooling-tower profiles, comet trajectories, LORAN navigation.

Summary

All four conics come from cutting a cone with a plane at different angles.
Eccentricity identifies them: e = 0 circle, 0 < e < 1 ellipse, e = 1 parabola, e > 1 hyperbola.
Ellipse and hyperbola differ only by a sign: use c² = a² − b² for the ellipse, c² = a² + b² for the hyperbola.
Always rewrite an equation in standard form before reading off the centre and the coefficients.