8.1 Conic Sections

Overview of Conic Sections

Conic sections (or conics) are the curves obtained by intersecting a plane with a double cone. Depending on the angle and position of the plane, four main types of curves appear: circle, parabola, ellipse, and hyperbola. In this chapter we describe what these curves have in common, how to recognize them algebraically, and what geometric features they share. Each specific conic type will be studied more closely in its own chapter.

Geometric Origin

Imagine a right circular double cone: two identical cones placed tip to tip. A conic section is the intersection of this double cone with a plane.

Different positions of the cutting plane produce different curves:

• A plane cutting the cone perpendicular to its axis gives a circle.
• A plane cutting at a slant, but not steep enough to meet both halves of the cone, gives an ellipse.
• A plane cutting parallel to a generator (slant edge) of the cone gives a parabola.
• A plane cutting both halves of the double cone gives a hyperbola.

In all cases, the curve is the set of all points where the plane and the cone meet.

Algebraic View: Quadratic Equations in Two Variables

Every conic section in the plane can be described by a second-degree equation in $x$ and $y$ of the general form

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

where $A,B,C,D,E,F$ are real constants and at least one of $A,B,C$ is nonzero.

This is called a general quadratic equation in two variables. Conic sections are exactly the curves whose equations can be written in this form, possibly after shifting or rotating the coordinate system.

Later chapters (Circles, Parabolas, Ellipses, Hyperbolas) will focus on simplified, standard forms of these equations and show how to convert between general and standard forms.

Classification by Coefficients

When the $xy$ term is absent ($B = 0$) and the coordinate axes are not rotated, you can often classify a conic just by looking at the coefficients of $x^2$ and $y^2$:

• Circle: $A = C \ne 0$, and $B = 0$.
• Ellipse (not a circle): $A$ and $C$ have the same sign, but $A \ne C$, and $B = 0$.
• Parabola: Exactly one of $A$ or $C$ is nonzero (the other is $0$), and $B = 0$.
• Hyperbola: $A$ and $C$ have opposite signs, and $B = 0$.

If $B \ne 0$, the axes of the conic are generally rotated relative to the $x$- and $y$-axes. Classification is then more subtle and typically uses linear algebra techniques (diagonalization of quadratic forms). That level of detail is beyond this introductory chapter.

It is also possible for the quadratic equation to represent degenerate cases, such as:

• a single point,
• a line or pair of lines,
• or the empty set (no real points satisfying the equation).

These are sometimes called degenerate conics and usually occur when the cone or the plane is treated in a limiting way (for example, the plane just touches the cone at its vertex).

Focus–Directrix Definition

Besides the cone-cutting picture and the algebraic description, conic sections can also be defined via distances to a point and a line in the plane.

Let:

• $F$ be a fixed point in the plane (the focus).
• $l$ be a fixed line, not passing through $F$ (the directrix).
• $e$ be a fixed positive constant called the eccentricity.

A conic section is the set of all points $P$ in the plane such that

$$
\frac{\text{distance from }P\text{ to }F}{\text{distance from }P\text{ to }l} = e.
$$

Different values of the eccentricity $e$ produce different types of conics:

• $e = 1$: the conic is a parabola.
• $0 < e < 1$: the conic is an ellipse.
• $e > 1$: the conic is a hyperbola.

A circle can be viewed as a special case of an ellipse with eccentricity $0$, but in the focus–directrix description it is usually treated separately (using two foci that coincide).

This unified definition emphasizes the geometric similarity among conics rather than how they arise from a cone.

Common Geometric Features

Although circles, parabolas, ellipses, and hyperbolas have distinct shapes, they share many geometric ideas. The later chapters on each type will make these ideas concrete; here we highlight the general concepts that recur.

Center and Symmetry

Many conics have a center: a point about which the conic is symmetric.

• A circle has a center, obvious from its definition.
• An ellipse has a center at the midpoint of its major and minor axes.
• A hyperbola has a center midway between its two branches.
• A parabola does not have a center of symmetry in the same sense, but it has an axis of symmetry: a line that reflects the curve onto itself.

Symmetry is important for:

• sketching the curve quickly,
• understanding its standard equation,
• simplifying distance and area calculations.

Axes

Conics often have special lines called axes:

• For a circle, every line through the center can be viewed as an axis of symmetry.
• For an ellipse, there are two main axes:
• the major axis (longest diameter),
• the minor axis (shortest diameter).
• For a hyperbola, each branch has a symmetry axis, and these axes intersect at the center.
• For a parabola, the axis of symmetry passes through the vertex and focus.

In standard positions, these axes are usually aligned with the coordinate axes, which simplifies their equations. When the $xy$ term is present, the axes of the conic are rotated relative to the $x$- and $y$-axes.

Foci and Directrices

Except for the circle (which can be treated as a limiting ellipse), the standard conics have:

• One focus and one directrix in the case of a parabola.
• Two foci and associated directrices in the case of an ellipse.
• Two foci and associated directrices in the case of a hyperbola.

These objects control how “stretched” the conic is. For example, in an ellipse the sum of distances to the two foci is constant; in a hyperbola the difference of distances to the two foci is constant.

The relative position of the focus and directrix reflects the eccentricity $e$.

Vertices

A vertex is a point where the conic meets one of its principal axes in an extreme way:

• For a parabola, the vertex is the “turning point”; it is midway between focus and directrix.
• For an ellipse, the vertices lie at the ends of the major axis.
• For a hyperbola, the vertices are the closest points on each branch to the center, lying along the transverse axis.
• A circle does not usually use the term vertex, but you can think of any point where a diameter meets the circle as an “extreme” point in one direction.

Vertices are typically easy to read off from standard equations and are extremely useful for graphing.

Transformations and Standard Position

Conic sections can appear in many different orientations and positions in the plane. However, by applying simple geometric transformations, you can often bring a given conic into a standard position where its equation is simpler and its properties are more obvious.

The key transformations are:

• Translations: shifting the graph left/right and up/down. Algebraically, this corresponds to replacing $x$ by $x - h$ and $y$ by $y - k$ for some constants $h,k$. This can move the center or vertex of the conic to a convenient point, often the origin.
• Reflections: flipping the graph across an axis or line, which changes signs of $x$ or $y$ (or both) in the equation.
• Rotations: turning the coordinate axes by some angle around the origin. This can eliminate the $xy$ term in many quadratic equations and align the conic’s axes with the new coordinate axes.

In practice, for many problems you will work mainly with conics whose axes are already aligned with the $x$- and $y$-axes and whose centers or vertices are at the origin or at easily found points. More complicated orientations are often handled with linear algebra methods.

Degenerate Conics

Not every quadratic equation in two variables represents a “full” circle, ellipse, parabola, or hyperbola. Certain parameter choices cause the conic to “collapse” into a simpler figure. Some common degenerate cases include:

• A point: e.g. an ellipse that has shrunk until both axes length are $0$.
• A line or pair of intersecting lines: e.g. $x^2 - y^2 = 0$ factors as $(x - y)(x + y) = 0$.
• The empty set: no real $(x,y)$ satisfy the equation.

These degenerate conics still fit into the algebraic framework of quadratic equations, but geometrically they are limiting cases.

Applications of Conic Sections

Conic sections are not just abstract curves; they model many real-world phenomena.

Some important applications include:

• Optics and reflection:
• Parabolic mirrors focus parallel light rays to a single point (the focus).
• Elliptical mirrors have the property that light from one focus reflects toward the other focus.
• Planetary motion:
• Under Newtonian gravity, the orbits of planets and satellites are conic sections (ellipses, parabolas, or hyperbolas), with the central body at a focus.
• Engineering and design:
• Bridges, arches, and antenna dishes often use parabolic or circular arcs for structural and functional reasons.
• Ellipses and circles appear naturally in cross-sections of cylinders, cones, and other 3D objects.
• Navigation and location:
• Some positioning methods use hyperbolas or circles defined by constant differences or sums of distances to known points.

Later chapters on each specific conic—circles, parabolas, ellipses, and hyperbolas—will explore their standard equations, graphs, and applications in more detail.