Conics Formulas- Complete Reference Guide
Conics Formulas: Complete Reference Guide
Conic sections are the curves you get when you slice a cone with a plane. Circle, ellipse, parabola, hyperbola—four shapes, four sets of rules. This guide gives you every formula you need, organized so you can find what you're looking for fast.
No fluff. Just the math.
The Four Conics at a Glance
| Conic | Shape | Eccentricity (e) |
|---|---|---|
| Circle | Perfect round | e = 0 |
| Ellipse | Stretched circle | 0 < e < 1 |
| Parabola | U-shaped curve | e = 1 |
| Hyperbola | Two separate curves | e > 1 |
Circle Formulas
A circle is a special case of an ellipse where all points are equidistant from the center. The distance is the radius.
Standard Equation (Center at Origin)
x² + y² = r²
Where r is the radius. That's it.
Standard Equation (Center at (h, k))
(x - h)² + (y - k)² = r²
When the center isn't at the origin, shift everything by (h, k).
Circle Formulas Quick Reference
- Radius: r = √(x² + y²) from center (origin)
- Diameter: d = 2r
- Circumference: C = 2πr
- Area: A = πr²
Ellipse Formulas
An ellipse is stretched from a circle. It has two radii: the semi-major axis (a) and semi-minor axis (b).
Standard Equation (Center at Origin, Horizontal Major Axis)
x²/a² + y²/b² = 1
When a > b, the major axis is horizontal. When b > a, it's vertical.
Standard Equation (Center at Origin, Vertical Major Axis)
x²/b² + y²/a² = 1
Ellipse with Center (h, k)
(x - h)²/a² + (y - k)²/b² = 1
Ellipse Key Measurements
- Semi-major axis: a (half the longest diameter)
- Semi-minor axis: b (half the shortest diameter)
- Distance from center to focus: c = √(a² - b²)
- Area: A = πab
- Perimeter (approximation): P ≈ π[3(a + b) - √((3a + b)(a + 3b))]
- Eccentricity: e = c/a
- Focal length: 2c
Parabola Formulas
A parabola is the set of all points equidistant from a focus point and a directrix line. It has one axis of symmetry.
Standard Equation (Vertex at Origin, Opens Up/Down)
y² = 4px opens right when p > 0, left when p < 0
x² = 4py opens up when p > 0, down when p < 0
Parabola with Vertex (h, k)
(y - k)² = 4p(x - h) (horizontal axis)
(x - h)² = 4p(y - k) (vertical axis)
Parabola Key Measurements
- Vertex: (h, k) — the lowest or highest point
- Focus: distance p from vertex along axis of symmetry
- Directrix: line perpendicular to axis, distance p from vertex
- Focal length: |p|
- Latus rectum length: |4p|
Hyperbola Formulas
A hyperbola has two separate branches. It has two foci and two vertices. The difference in distances to the foci is constant.
Standard Equation (Center at Origin, Opens Left/Right)
x²/a² - y²/b² = 1
Standard Equation (Center at Origin, Opens Up/Down)
y²/a² - x²/b² = 1
Hyperbola with Center (h, k)
(x - h)²/a² - (y - k)²/b² = 1 (horizontal)
(y - k)²/a² - (x - h)²/b² = 1 (vertical)
Hyperbola Key Measurements
- Transverse axis: 2a (distance between vertices)
- Conjugate axis: 2b
- Distance from center to focus: c = √(a² + b²)
- Eccentricity: e = c/a
- Asymptotes: y = ±(b/a)x for centered at origin
General Conic Equation
Every conic can be written in the general form:
Ax² + Bxy + Cy² + Dx + Ey + F = 0
The discriminant (B² - 4AC) tells you which conic you have:
| Value of B² - 4AC | Conic Type |
|---|---|
| < 0 | Ellipse (or circle if A = C and B = 0) |
| = 0 | Parabola |
| > 0 | Hyperbola |
Conic Formulas Comparison Table
| Property | Circle | Ellipse | Parabola | Hyperbola |
|---|---|---|---|---|
| Standard form | x² + y² = r² | x²/a² + y²/b² = 1 | y² = 4px | x²/a² - y²/b² = 1 |
| Eccentricity | 0 | 0 < e < 1 | 1 | e > 1 |
| Foci | 1 (center) | 2 | 1 | 2 |
| Directrix | None | 2 | 1 | 2 |
| Area formula | πr² | πab | N/A | N/A |
How to Identify a Conic from an Equation
Follow these steps when you see an equation and need to classify it:
Step 1: Check for xy Term
If there's an xy term, rotate your axes. The angle of rotation is:
θ = ½ arctan(B/A)
Step 2: Check the Discriminant
Calculate B² - 4AC from the general form.
Negative = ellipse, Zero = parabola, Positive = hyperbola.
Step 3: Check Coefficients
If A = C and B = 0, it's a circle.
If A or C is zero but not both, it's a parabola.
How to Write the Equation of a Conic
Writing a Circle Equation
Example: Circle with center (3, -2) and radius 5.
Plug into (x - h)² + (y - k)² = r²:
(x - 3)² + (y + 2)² = 25
Writing an Ellipse Equation
Example: Ellipse with center (0, 0), a = 5, b = 3, horizontal major axis.
x²/25 + y²/9 = 1
Writing a Parabola Equation
Example: Parabola with vertex (0, 0), focus at (0, 4).
p = 4, opens up, so x² = 4(4)y:
x² = 16y
Writing a Hyperbola Equation
Example: Hyperbola with center (0, 0), a = 3, b = 4, opens left/right.
x²/9 - y²/16 = 1
Distance and Focus Formulas
The definition-based formulas for each conic:
- Circle: Distance from center = radius
- Ellipse: Distance to both foci sums to 2a
- Parabola: Distance to focus = distance to directrix
- Hyperbola: Distance to both foci differs by 2a
Parametric Equations
Sometimes parametric form is more useful:
Circle
x = r cos(t), y = r sin(t)
Ellipse
x = a cos(t), y = b sin(t)
Parabola
x = t, y = t² (vertex at origin, opens up)
Hyperbola
x = a sec(t), y = b tan(t)
Quick Reference: All Conics Summary
| Conic | Centered at Origin | With Center (h, k) | Key Feature |
|---|---|---|---|
| Circle | x² + y² = r² | (x-h)² + (y-k)² = r² | All points = r from center |
| Ellipse | x²/a² + y²/b² = 1 | (x-h)²/a² + (y-k)²/b² = 1 | Sum of distances to foci = 2a |
| Parabola | y² = 4px | (y-k)² = 4p(x-h) | Distance to focus = distance to directrix |
| Hyperbola | x²/a² - y²/b² = 1 | (x-h)²/a² - (y-k)²/b² = 1 | Difference of distances to foci = 2a |
Bookmark this page. You'll be back.