Conic sections

Slice a party hat (a cone) with a flat plane at different angles and you get, in turn, a circle, an oval, an open U-shaped curve, or a curve with two mirror-image branches. These four — circle, ellipse, parabola, hyperbola — are the conic sections, and this article is the map that connects them before ISC Class 11 studies the parabola and ellipse in full algebraic detail. The four curves look unrelated, but a single ratio unifies them: how far a point on the curve sits from a fixed focus compared to a fixed directrix line. That one number, the eccentricity $e$ , tells you instantly which curve you are looking at, and it is the thread every later formula in this unit hangs from.

The core insight

Every conic section is the locus of a point whose distance from a fixed focus stays in a constant ratio $e$ (the eccentricity) to its distance from a fixed directrix line, and the value of $e$ alone — less than, equal to, or greater than $1$ — decides whether the curve is an ellipse, a parabola, or a hyperbola.

The core idea

A conic section is literally the curve you get where a plane meets a double (two-nappe) right circular cone. The angle of the cut relative to the cone's axis decides which of four shapes appears.

Circle — the plane is perpendicular to the axis, cutting straight across one nappe.
Ellipse — the plane cuts one nappe on a slant, steeper than the cone's own slant edge but not perpendicular to the axis.
Parabola — the plane runs exactly parallel to one generating (slant) edge of the cone.
Hyperbola — the plane is steep enough to slice through both nappes at once, giving two separate branches.

This geometric picture is intuitive but clumsy to compute with. Algebra needs a flat, two-dimensional recipe, and every conic obeys one: it is the set of points $P$ for which the distance to a fixed point (the focus) divided by the distance to a fixed line (the directrix) is a constant, the eccentricity $e$ :

\frac{PF}{PM} = e

where: $PF$ is the distance from a point $P$ on the curve to the focus $F$ , $PM$ is the perpendicular distance from $P$ to the directrix line, and $e$ is a fixed positive number that does not change as $P$ moves along the curve. The value of $e$ alone classifies the curve:

$e = 0$ : circle, a limiting case normally handled directly by centre and radius rather than this ratio.
$0 < e < 1$ : ellipse.
$e = 1$ : parabola — focus and directrix are equally weighted.
$e > 1$ : hyperbola.

Degenerate conics. If the cutting plane happens to pass exactly through the cone's apex, the smooth curve collapses into a single point, a single line, or a pair of intersecting lines. These are edge cases worth recognising but are not the target of the standard-equation work ahead. The figure below lines up the two pictures — the solid-geometry cut and the resulting plane curve — for all four cases.

Figure

A double cone sliced by planes at four different angles producing a circle an ellipse a parabola and a hyperbola

Notice how the compulsory Section A "Circles" topic already sits inside this picture as the $e = 0$ limit, even though that chapter never mentions eccentricity or a directrix — it works purely with centre and radius. The moment a problem starts talking about eccentricity, focus, or directrix, it has stepped into the optional Section B "Conic Section" unit that this article, and the Parabola and Ellipse articles that follow it, belong to.

Derivation ADeriving the general conic equation from a focus and a directrix

The goal is to convert the geometric ratio $PF/PM = e$ into an algebraic equation in $x$ and $y$ , and to show that this equation is always second degree — a quadratic in the coordinates — which is exactly why every conic section, regardless of its shape, can be written using $x^2$ , $y^2$ , and $xy$ terms, and why the coefficient pattern shifts in a very specific way as $e$ crosses the value $1$ .

Step 1. Set up convenient coordinates. Place the focus at the origin, $F = (0,0)$ , and take the directrix as the vertical line $x = -k$ for some fixed $k > 0$ . Let $P = (x, y)$ be any point on the conic.

Step 2. Write both distances. The distance from $P$ to the focus is $PF = \sqrt{x^2 + y^2}$ . The perpendicular distance from $P$ to the vertical line $x = -k$ is simply $PM = x + k$ (taking $P$ to the right of the directrix, so this stays positive).

Step 3. Impose the defining ratio and square both sides to clear the square root. Since $PF = e \cdot PM$ , we get $\sqrt{x^2+y^2} = e(x+k)$ , and squaring gives $x^2 + y^2 = e^2(x+k)^2 = e^2x^2 + 2e^2kx + e^2k^2$ .

Step 4. Collect everything on one side. Rearranging, $(1 - e^2)\,x^2 + y^2 - 2e^2k\,x - e^2k^2 = 0$ . This is a genuine second-degree equation in $x$ and $y$ for any value of $e$ — confirming algebraically that "conic section" and "curve with a quadratic equation" describe the same family. Read the coefficient of $x^2$ : when $e < 1$ it is positive, when $e > 1$ it is negative (opposite sign to the $y^2$ term), and when $e = 1$ it vanishes entirely, leaving an equation linear in $x$ and quadratic only in $y$ — the signature shape of a parabola. This single coefficient, $1 - e^2$ , is the algebraic seed from which the separate standard-form equations of the ellipse, parabola, and hyperbola are built in the next two articles.

Worked example

Classify the conic whose points satisfy $\dfrac{PF}{PM} = \dfrac{2}{3}$ , and confirm the shape of its general equation.

Step 1. Read off the eccentricity. Here $e = \dfrac{2}{3}$ , and since $0 < \dfrac{2}{3} < 1$ , the curve is an ellipse.

Step 2. Check the coefficient pattern from the derivation above. With $e = \dfrac23$ , $e^2 = \dfrac49$ , so $1 - e^2 = \dfrac59$ , a positive number. The general equation $(1-e^2)x^2 + y^2 - 2e^2kx - e^2k^2 = 0$ therefore has $x^2$ and $y^2$ both with positive coefficients — exactly the pattern an ellipse's equation should show, confirming the classification without needing to plot a single point.

Step 3. Contrast with the boundary case. If instead $e = 1$ (focus and directrix distances always equal), then $1 - e^2 = 0$ , the $x^2$ term disappears, and the shape is a parabola — matching Worked example 2 of the companion Parabola article, where "equidistant from a fixed point and a fixed line" is precisely how the parabola is defined. Answer: $e = \dfrac23$ gives an ellipse, with the general equation's $x^2$ -coefficient $1-e^2 = \dfrac59 > 0$ confirming it.

Common traps

Treating conic sections as compulsory content. They live in the optional Section B — a student examined under Section C never sees eccentricity, parabolas, or ellipses on the paper.
Confusing the 3-D cone-cutting picture with the 2-D focus-directrix definition. Both describe the same curves, but every calculation uses the algebraic ratio $PF/PM = e$ , never the solid cone directly.
Mixing up which inequality on $e$ gives which curve. Anchor $e = 1$ (parabola) first, then reason outward: less than $1$ is more closed (ellipse), greater than $1$ is more open (hyperbola).
Forgetting that $e = 0$ is a limit, not a working formula. Circles use the centre-radius definition from the compulsory Coordinate Geometry chapter, not the eccentricity ratio.

Ask the tutor: If you want to see how this general equation turns into the specific standard forms for the parabola or ellipse, or want to check whether your school covers Section B, ask the AI tutor before moving on.

Why do the circle and the ellipse both come from cutting one nappe, but look so different

The difference is the angle of the cut relative to the cone's axis, not which nappe is involved. A plane exactly perpendicular to the axis meets the cone in a curve where every point is the same distance from the centre — a circle, $e = 0$ . Tilt that same plane away from perpendicular, and the curve stretches into an oval whose distance from the centre now varies smoothly with direction — an ellipse, $0 < e < 1$ . The circle is really just the ellipse's most symmetric special case.

Is it worth memorising the general second-degree equation from the derivation, or just the classification by e

For ISC Class 11 exam purposes, the classification by $e$ (less than, equal to, or greater than $1$ ) is what gets tested directly, and that is what you should have instantly ready. The general equation is worth understanding once, because it explains why the classification works and previews the standard-form equations you will meet for the parabola and ellipse — but you will not be asked to re-derive it from scratch in a typical problem.

If a plane passes through the cone's apex, why isn't that also called a conic section

It technically is called a degenerate conic, but it fails to produce a smooth curve — instead you get a single point, a single line, or two intersecting lines, depending on the exact angle. These cases are mentioned for completeness because they sit at the same boundary as the four main curves, but the standard-equation work in this unit is built entirely around the smooth, non-degenerate cases: circle, ellipse, parabola, hyperbola.

Conic sections

The core idea

Worked example

Common traps

Before this

Where this leads