A conic section (or conic) is a curve obtained by intersecting a plane with a double-napped cone. The shape of the curve depends on the angle of the plane.
A conic can also be defined in 2D as the locus of a point P that moves so that the ratio of its distance from a fixed point (the focus) to its perpendicular distance from a fixed line (the directrix) is a constant value e, called the eccentricity.
Distance from Focus / Distance from Directrix = e
e = 1, the conic is a Parabola.e < 1, the conic is an Ellipse.e > 1, the conic is a Hyperbola.e = 0, it is a Circle (a special case of an ellipse).A parabola is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).
[Image of a parabola, showing the focus, directrix, vertex, and axis of symmetry]The "various forms" refer to the four standard orientations of a parabola with its vertex at the origin (0, 0).
| Property | y² = 4ax(Opens Right) |
y² = -4ax(Opens Left) |
x² = 4ay(Opens Up) |
x² = -4ay(Opens Down) |
|---|---|---|---|---|
| Vertex | (0, 0) | (0, 0) | (0, 0) | (0, 0) |
| Focus | (a, 0) | (-a, 0) | (0, a) | (0, -a) |
| Directrix | x = -a | x = a | y = -a | y = a |
| Axis | y = 0 (x-axis) | y = 0 (x-axis) | x = 0 (y-axis) | x = 0 (y-axis) |
| Latus Rectum | 4a | 4a | 4a | 4a |
4a.An ellipse is the set of all points P such that the sum of the distances from P to two fixed points (the foci, F₁ and F₂) is a constant (2a).
The standard forms have the center at the origin (0, 0).
Relationship between a, b, and e: b² = a²(1 - e²) and e² = (a² - b²) / a²
| Property | x²/a² + y²/b² = 1(Horizontal, a > b) |
x²/b² + y²/a² = 1(Vertical, a > b) |
|---|---|---|
| Center | (0, 0) | (0, 0) |
| Major Axis | Horizontal, length 2a | Vertical, length 2a |
| Minor Axis | Vertical, length 2b | Horizontal, length 2b |
| Vertices | (±a, 0) | (0, ±a) |
| Foci | (±ae, 0) or (±c, 0) | (0, ±ae) or (0, ±c) |
| Eccentricity | e = c/a < 1 |
e = c/a < 1 |
| Latus Rectum | 2b²/a | 2b²/a |
a² is under x², it's a horizontal ellipse. If a² is under y², it's a vertical ellipse.
A hyperbola is the set of all points P such that the *difference* of the distances from P to two fixed points (the foci, F₁ and F₂) is a constant (2a).
The standard forms have the center at the origin (0, 0).
Relationship between a, b, and e: b² = a²(e² - 1) and e² = (a² + b²) / a²
| Property | x²/a² - y²/b² = 1(Horizontal) |
y²/a² - x²/b² = 1(Vertical) |
|---|---|---|
| Center | (0, 0) | (0, 0) |
| Transverse Axis | Horizontal, length 2a | Vertical, length 2a |
| Conjugate Axis | Vertical, length 2b | Horizontal, length 2b |
| Vertices | (±a, 0) | (0, ±a) |
| Foci | (±ae, 0) or (±c, 0) | (0, ±ae) or (0, ±c) |
| Eccentricity | e = c/a > 1 |
e = c/a > 1 |
| Latus Rectum | 2b²/a | 2b²/a |
| Asymptotes | y = ±(b/a)x | y = ±(a/b)x |
x² is positive, it's a horizontal hyperbola. If y² is positive, it's a vertical hyperbola.
e = 1 → Parabola (distance to focus = distance to directrix)e < 1 → Ellipse (distance to focus < distance to directrix)e > 1 → Hyperbola (distance to focus > distance to directrix)y² = 4ax. All others are variations.x²/a² + y²/b² = 1 (a 'plus' sign). 'a' is always the larger denominator.x²/a² - y²/b² = 1 (a 'minus' sign). 'a' is always with the positive term.a² = b² + c².c² = a² + b².