One cone, four curves
A conic section is obtained when a plane cuts a double cone. The angle and position of the plane determine the curve:
a circle appears when the plane is perpendicular to the cone's axis,
an ellipse appears when the plane cuts one nappe at an angle without becoming parallel to a generator,
a parabola appears when the plane is parallel to a generator,
a hyperbola appears when the plane cuts both nappes.
Each conic can also be described using a fixed point called a focus and a fixed line called a directrix. For a point P on the conic, the ratio
distance from P to focus / distance from P to directrix = e
is constant. The constant e is the eccentricity.
Conic | Eccentricity |
|---|---|
Circle | e = 0 |
Ellipse | 0 < e < 1 |
Parabola | e = 1 |
Hyperbola | e > 1 |
Circle
A circle with centre (h, k) and radius r has equation:
(x − h)² + (y − k)² = r²
The special case with centre at the origin is x² + y² = r².
A general equation x² + y² + 2gx + 2fy + c = 0 represents a circle with centre (−g, −f) and radius √(g² + f² − c), provided the expression under the square root is positive.
Example. Complete squares in x² + y² − 6x + 4y − 12 = 0:
(x − 3)² + (y + 2)² = 25.
The centre is (3, −2) and the radius is 5.
Parabola
A parabola is the set of points equidistant from a focus and a directrix. Its eccentricity is 1.
Standard equation | Opens toward | Focus | Directrix |
|---|---|---|---|
y² = 4ax | positive x-axis | (a, 0) | x = −a |
y² = −4ax | negative x-axis | (−a, 0) | x = a |
x² = 4ay | positive y-axis | (0, a) | y = −a |
x² = −4ay | negative y-axis | (0, −a) | y = a |
For all four forms, the vertex is the origin and the length of the latus rectum is 4a.
Example. In y² = 12x, 4a = 12, so a = 3. The focus is (3, 0), the directrix is x = −3, and the latus rectum has length 12.
Ellipse
An ellipse is the set of points for which the sum of distances from two fixed foci is constant. For an ellipse centred at the origin with major axis along the x-axis:
x²/a² + y²/b² = 1, where a > b > 0.
Its vertices are (±a, 0), its co-vertices are (0, ±b), and its foci are (±c, 0), where:
c² = a² − b² and e = c/a.
The major axis has length 2a, the minor axis has length 2b, and each latus rectum has length 2b²/a.
If the larger denominator is under y², the major axis and foci lie along the y-axis.
Example. For x²/25 + y²/9 = 1, a = 5, b = 3 and c = √(25 − 9) = 4. The foci are (±4, 0) and e = 4/5.
Hyperbola
A hyperbola is the set of points for which the absolute difference of distances from two fixed foci is constant. For a hyperbola centred at the origin opening left and right:
x²/a² − y²/b² = 1.
Its vertices are (±a, 0), and its foci are (±c, 0), where:
c² = a² + b² and e = c/a > 1.
Its asymptotes are y = ±(b/a)x. The curve approaches these lines but never meets them.
For y²/a² − x²/b² = 1, the hyperbola opens upward and downward, with vertices and foci on the y-axis.
Example. In x²/16 − y²/9 = 1, a = 4, b = 3 and c = 5. The foci are (±5, 0), eccentricity is 5/4, and the asymptotes are y = ±(3/4)x.
How to identify the conic
After moving constants and completing squares when needed, examine the squared terms:
same sign and same coefficient: circle,
same sign and different coefficients: ellipse,
only one variable squared: parabola,
opposite signs: hyperbola.
Always rewrite the equation so that its right side is 1 before reading a² and b² from an ellipse or hyperbola.
Summary
The standard equation reveals a conic's orientation and key measurements.
For an ellipse, c² = a² − b² and e < 1.
For a hyperbola, c² = a² + b² and e > 1.
For a parabola, the coefficient 4a locates the focus and directrix.