MTH61: CONIC SECTION

CONIC SECTION

A conic section, conic or a quadratic curve is a curve obtained from a cone’s surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type.

The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga’s systematic work on their properties.

The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions.

One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a focus, and some particular line, called a directrix, are in a fixed ratio, called the eccentricity.

The type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2; that is, as the set of points whose coordinates satisfy a quadratic equation in two variables which can be written in the form ��2+���+��2+��+��+�=0. The geometric properties of the conic can be deduced from its equation.

In the Euclidean plane, the three types of conic sections appear quite different, but share many properties. By extending the Euclidean plane to include a line at infinity, obtaining a projective plane, the apparent difference vanishes: the branches of a hyperbola meet in two points at infinity, making it a single closed curve; and the two ends of a parabola meet to make it a closed curve tangent to the line at infinity. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically.

Cones

Circle
straight through

Ellipse
slight angle

Parabola
parallel to edge of cone

Hyperbola
steep angle

So all those curves are related!

Focus!

The curves can also be defined using a straight line and a point (called the directrix and focus).

When we measure the distance:

• from the focus to a point on the curve, and
• perpendicularly from the directrix to that point

the two distances will always be the same ratio.

• For an ellipse, the ratio is less than 1
• For a parabola, the ratio is 1, so the two distances are equal.
• For a hyperbola, the ratio is greater than 1

Eccentricity

That ratio above is called the “eccentricity“, so we can say that any conic section is:

For:

• 0 < eccentricity < 1 we get an ellipse,
• eccentricity = 1 a parabola, and
• eccentricity > 1 a hyperbola.

A circle has an eccentricity of zero, so the eccentricity shows us how “un-circular” the curve is. The bigger the eccentricity, the less curved it is.

Latus Rectum

The latus rectum (no, it is not a rude word!) runs parallel to the directrix and passes through the focus. Its length:

• In a parabola, is four times the focal length
• In a circle, is the diameter
• In an ellipse, is 2b²/a (where a and b are one half of the major and minor diameter).

Here is the major axis and minor axis of an ellipse.

There is a focus and directrix on each side (ie a pair of them).

Equations

When placed like this on an x-y graph, the equation for an ellipse is:

x²a² + y²b² = 1

The special case of a circle (where radius=a=b):

x²a² + y²a² = 1

And for a hyperbola it is:

x²a² − y²b² = 1

General Equation

We can make an equation that covers all these curves.

Because they are plane curves (even though cut out of the solid) we only have to deal with Cartesian (“x” and “y”) Coordinates.

But these are not straight lines, so just “x” and “y” will not do … we need to go to the next level, and have:

• x² and y²,
• and also x (without y), y (without x),
• x and y together (xy)
• and a constant term.

There, that should do it!

And each one needs a factor (A,B,C etc) …

So the general equation that covers all conic sections is:

And from that equation we can create equations for the circle, ellipse, parabola and hyperbola.

PARABOLA

ELLIPSE

HYPERBOLA