Conic Sections: Types, Formulas & Equations

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, $e$, 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 $e$ — the constant ratio of distance-from-focus to distance-from-directrix. As $e$ 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 $e$ 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 $e$, 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.

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 $c^2 = a^2 - b^2$. In the parabola $y^2 = 4ax$, 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 $c^2 = a^2 + b^2$ — note the plus sign, the single change from the ellipse.

The two relations are worth pausing on, because the only difference between an ellipse and a hyperbola in this notation is one sign: $c^2 = a^2 - b^2$ for the ellipse, $c^2 = a^2 + b^2$ for the hyperbola.

How Do You Identify a Conic From Its Equation?

Any conic can also be written in the general second-degree form:

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.$$

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.

So before any algebra, the signs and relative sizes of the $x^2$ and $y^2$ coefficients already name the curve.

Examples of Conic Section

With the four types, their eccentricities, and the standard equations in hand, here are the curves doing real work. The problems move from naming a conic to finding its eccentricity.

Example 1: Identify the conic: $x^2 + y^2 = 25$

Both squared terms have coefficient $1$ (so $A = C$) and the same sign. That is the circle test. Here $r^2 = 25$, so $r = 5$.

Final answer: a circle of radius 5.

Example 2: Identify the conic from $4x^2 + 9y^2 = 36$, and a student answers "circle, because both terms are positive"

Check the test more carefully. A circle needs the $x^2$ and $y^2$ coefficients to be equal, not just both positive. Here they are $4$ and $9$, same sign but unequal, which is the ellipse test, not the circle test.

Divide through by $36$ to put it in standard form:

$$\frac{4x^2}{36} + \frac{9y^2}{36} = 1 ;\Rightarrow; \frac{x^2}{9} + \frac{y^2}{4} = 1.$$

So $a^2 = 9$, $b^2 = 4$.

Final answer: an ellipse with $a = 3$, $b = 2$.

Example 3: Find the eccentricity of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$

Here $a^2 = 25$, $b^2 = 16$, so $c^2 = a^2 - b^2 = 25 - 16 = 9$, giving $c = 3$. Eccentricity is $e = \tfrac{c}{a} = \tfrac{3}{5} = 0.6$. Since $0 < 0.6 < 1$, it is indeed an ellipse.

Final answer: $e = 0.6$.

Example 4: Identify the conic and its features: $y^2 = 16x$

Only $y$ is squared (there is no $x^2$ term), which is the parabola test. Comparing with $y^2 = 4ax$ 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 $\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1$

For a hyperbola, $c^2 = a^2 + b^2 = 9 + 16 = 25$, so $c = 5$, with $a = 3$. Then $e = \tfrac{c}{a} = \tfrac{5}{3} \approx 1.67$. Since $e > 1$, it is a hyperbola.

Final answer: $e = \tfrac{5}{3} \approx 1.67$.

Example 6: A satellite dish has a parabolic cross-section $y^2 = 8x$ (units in metres). How far from the vertex should the receiver sit?

The receiver goes at the focus, because a parabola reflects all incoming parallel rays to that single point. Comparing $y^2 = 8x$ with $y^2 = 4ax$ 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. Example 6 is this fact in action.

• 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.

This is also the topic where geometry and algebra finally fuse: a shape sliced from a cone in 3D becomes a tidy equation on a 2D grid. For a Grade 11 student, conic sections are the bridge into coordinate geometry, calculus, and physics all at once.

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.

Don't do this: Treat "both terms positive" as the circle test.

The correct way: A circle needs the $x^2$ and $y^2$ coefficients equal (and same sign). If they are unequal, it is an ellipse. Compare the numbers, not just the signs.

Mistake 2: Confusing the ellipse and hyperbola sign relations

Where it slips in: Computing $c$, the student uses $c^2 = a^2 - b^2$ for a hyperbola or $c^2 = a^2 + b^2$ for an ellipse.

Don't do this: Use the same $a$–$b$ relation for both.

The correct way: Ellipse uses minus ($c^2 = a^2 - b^2$); hyperbola uses plus ($c^2 = a^2 + b^2$). The second-guesser who memorised one relation and applies it to both curves lands here — tie the sign to the curve's own equation sign (ellipse adds the fractions, hyperbola subtracts them).

Mistake 3: Forgetting the eccentricity boundaries

Where it slips in: A student computes $e$ correctly but then names the wrong curve.

Don't do this: Treat any $e$ between $0$ and $2$ as "probably an ellipse."

The correct way: The cut-offs are exact: $e = 0$ circle, $0 < e < 1$ ellipse, $e = 1$ parabola, $e > 1$ hyperbola. The boundary values $0$ and $1$ are not approximate.

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.

• Eccentricity $e$ alone sorts them: $0$ (circle), $0$–$1$ (ellipse), $1$ (parabola), $>1$ (hyperbola).

• Standard equations: circle $x^2+y^2=r^2$, ellipse $\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2}=1$, parabola $y^2=4ax$, hyperbola $\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}=1$.

• The ellipse uses $c^2 = a^2 - b^2$; the hyperbola uses $c^2 = a^2 + b^2$ — only the sign differs.

• The most common mistake is calling an unequal-coefficient equation a circle, or swapping the ellipse and hyperbola sign relations.

Practice These Problems to Solidify Your Understanding

1. Identify the conic: $\dfrac{x^2}{16} + \dfrac{y^2}{16} = 1$.

2. Find the eccentricity of the ellipse $\dfrac{x^2}{169} + \dfrac{y^2}{144} = 1$.

3. Identify the conic and find its focus: $y^2 = 20x$.

Answer to Question 1: equal denominators mean $A = C$, so it is a circle of radius 4 (it can be written $x^2 + y^2 = 16$). Answer to Question 2: $c^2 = 169 - 144 = 25$, $c = 5$, $a = 13$, so $e = \tfrac{5}{13} \approx 0.38$. Answer to Question 3: $4a = 20$, $a = 5$, so it is a parabola with focus $(5, 0)$.

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Also Read:

• Geometry — Concepts, Formulas, Shapes & Topics Guide

• Parabola

• Ellipse

• Equation of a Circle

• Focus of a Parabola

• Directrix of a Parabola

• Foci of an Ellipse