Conic Sections: Types, Formulas & Equations
Conic Sections: Types, Formulas & Equations
TL;DR
A conic section is the curve you get when a flat plane slices through a cone, and tilting the slice produces exactly four shapes: the circle, ellipse, parabola, and hyperbola. This article covers the definition, the four types, their eccentricity values, the focus-directrix idea, standard equations, six worked examples, and the mistakes students make most.
What Is a Conic Section?
A conic section is a curve formed by the intersection of a flat plane with a double cone (two identical cones joined at their tips). Depending on the angle at which the plane cuts the cone, the intersection is a circle, ellipse, parabola, or hyperbola — these four are the conic sections.
There is a second, equivalent way to define them that does not mention a cone at all: a conic is the set of all points whose distance from a fixed point (the focus) and a fixed line (the directrix) keep a constant ratio. That ratio is the curve's eccentricity, and it alone decides which of the four shapes you get. Both definitions describe the same curves; the focus-directrix version is the one that powers the equations.
What Are the Four Types of Conic Sections?
Each type is fixed by a single number, its eccentricity — the constant ratio of distance-from-focus to distance-from-directrix. As eccentricity grows, the curve opens up.
- Circle (e=0). The plane cuts straight across the cone, level with the base. Every point is the same distance from the centre.
- Ellipse (0<e<1). The plane tilts; the closed curve stretches into an oval with two foci.
- Parabola (e=1). The plane runs parallel to the cone's slant side; the curve opens and never closes.
- Hyperbola (e>1). The plane is steep enough to cut both cones, giving two separate open branches.
The single number does all the sorting: it is exactly 0 for a circle, between 0 and 1 for an ellipse, exactly 1 for a parabola, and more than 1 for a hyperbola. Knowing the eccentricity, you know the shape.
Standard Equations of the Conic Sections
When each curve is centred neatly on the origin of a coordinate plane, its equation takes a clean standard form. The letters below are not decoration: a and b set the size and stretch, and they relate the foci to the curve.
| Conic | Eccentricity | Standard equation |
|---|---|---|
| Circle | 0 | x² + y² = r² |
| Ellipse | 0<e<1 | (\frac{x²}{a²} + \frac{y²}{b²} = 1) |
| Parabola | 1 | y² = 4ax |
| Hyperbola | >1 | (\frac{x²}{a²} - \frac{y²}{b²} = 1) |
Reading the variables: in the circle, r is the radius. In the ellipse, a is half the longer (major) axis and b is half the shorter (minor) axis, and the foci sit at distance c from the centre where c² = a² - b². In the parabola, y² = 4ax, the value a is the distance from the vertex to the focus. In the hyperbola, a and b set the branches and the foci satisfy c² = a² + b².
How Do You Identify a Conic From Its Equation?
Any conic can also be written in the general second-degree form:
Ax² + Bxy + Cy² + Dx + Ey + F = 0.
When there is no xy term (B=0), a quick rule on A and C tells you the type:
- A = C (same coefficients) → circle.
- A and C same sign but unequal → ellipse.
- A or C is zero (only one squared term) → parabola.
- A and C opposite signs → hyperbola.
Examples of Conic Section
Example 1: Identify the conic: x² + y² = 25
Both squared terms have coefficient 1 (so A = C) and the same sign. That is the circle test. Here r² = 25, so r = 5.
Final answer: a circle of radius 5.
Example 2: Identify the conic from 4x² + 9y² = 36
A student answers "circle, because both terms are positive". Check the test more carefully: A circle needs the x² and y² coefficients to be equal; here they are 4 and 9, same sign but unequal, which is the ellipse test. Divide through by 36 to put it in standard form:
(\frac{4x²}{36} + \frac{9y²}{36} = 1) ⇒ (\frac{x²}{9} + \frac{y²}{4} = 1).
Final answer: an ellipse with a=3, b=2.
Example 3: Find the eccentricity of the ellipse (\frac{x²}{25} + \frac{y²}{16} = 1)
Here a² = 25, b² = 16, so c² = a²−b² = 25−16 = 9, giving c = 3. Eccentricity is e = (\frac{c}{a} = \frac{3}{5} = 0.6). Since 0 < 0.6 < 1, it is indeed an ellipse.
Final answer: e=0.6.
Example 4: Identify the conic and its features: y² = 16x
Only y is squared (there is no x² term), which is the parabola test. Comparing with y² = 4ax gives 4a = 16, so a = 4: the focus is at (4,0) and the curve opens rightward.
Final answer: a parabola, focus (4,0).
Example 5: Find the eccentricity of the hyperbola (\frac{x²}{9} - \frac{y²}{16} = 1)
For a hyperbola, c² = a² + b² = 9 + 16 = 25, so c = 5, with a=3. Then e= (\frac{c}{a} = \frac{5}{3} ≈ 1.67). Since e > 1, it is a hyperbola.
Final answer: e ≈ 1.67.
Example 6: A satellite dish has a parabolic cross-section y² = 8x (units in metres). How far from the vertex should the receiver sit?
The receiver goes at the focus, comparing y²=8x with y²=4ax gives 4a=8, so a=2.
Final answer: 2 metres from the vertex.
Why Conic Sections Matter
These four curves are not a classroom curiosity; they are the shapes the universe and our machines keep choosing.
- Planetary orbits. Every planet travels around the Sun in an ellipse with the Sun at one focus — Johannes Kepler's first law. Comets follow ellipses, parabolas, or hyperbolas depending on whether they return.
- Reflectors and dishes. A parabola reflects all incoming parallel rays to a single focus, which is why satellite dishes, car headlights (run in reverse), and solar cookers are parabolic.
- Whispering galleries. An ellipse reflects sound from one focus straight to the other, so a whisper at one focus of an elliptical hall is heard clearly across the room.
- Navigation and tracking. Hyperbolas underpin LORAN and GPS-style positioning, where time differences in signals trace a hyperbola of possible locations.
Where Students Trip Up on Conic Sections
Mistake 1: Calling an ellipse a circle
Where it slips in: An equation has two positive squared terms, and the student announces a circle without checking whether the coefficients are equal.
Mistake 2: Confusing the ellipse and hyperbola sign relations
Where it slips in: Computing c, the student uses the wrong relation.
Mistake 3: Forgetting the eccentricity boundaries
Where it slips in: A student computes e correctly but then names the wrong curve.
Key Takeaways
- A conic section is a curve made by slicing a cone with a plane; the four types are the circle, ellipse, parabola, and hyperbola.
- The most common mistake is calling an unequal-coefficient equation a circle.