Although it is possible to give analytic or algebraic descriptions of tangency, the
geometrical description leads to more interesting results.
Definition
A line L is a tangent to the conic C at the point P if
- L and C meet only in P, and
- there is a line L' parallel to L which meets C at least twice.
Of course, since a line has a linear equation, and a conic has quadratic equation
a line and conic meet in at most two points, we can delete the "at least" in(2).
An intuitive justification for the inclusion of clause (2), with illustrations, appears
elsewhere. We will provide proofs here.
First, we show that tangency is an affine concept.
The Affine Tangency Theorem
If L is a tangent at the point P on a conic C, and t is an affine transformation,
then t(L) is a tangent at t(P) on the conic t(C).
proof
Now Theorem AC2 shows that we need only investigate the existence of tangents
to the standard affine conics. Indeed, by the
Affine Transitivity Theorem, we
need only look at a particular point on each of these!
If we look at the standard conics in euclidean geometry, it is easy to see that the
line perpendicular to the axis of symmetry at a point where it cuts the conic is a
tangent to the conic. If this line cuts the conic again, by symmetry there would be
a third meeting, contrary to a remark above. Also, it is easy to see that no other
line through this point satisfies clause (1). Thus
Results for the standard conics can be also established by algebraic methods.
The Tangent Theorem
If P is a point on the conic C, then there is a unique line L tangent to C at P.
For completeness, we will give a proof, though it follows quickly from the preceding
remarks.
Although this is a uniform result for all conics, there are differences between the
types of conics. These actually characterize the types. Suppose that L is a line
and C is a conic, then we can ask how many tangents to C are parallel to L?
Also, if there are several such tangents, how are the points of contacts related?
Definition
If L is a line, then <L> is the family of lines
parallel to L.
Since affine transformations preserve parallels, t(<L>) = <t(L)>.
Our results follow from those for the standard conics. These are established
The Ellipse-Tangent Theorem
If E is an ellipse and L is a line, then
(1) the family <L> contains exactly two tangents to E, and
(2) the tangents to E at P and Q are parallel if and only if PQ is a diameter.
proof
Part (2) is sometimes known as the Parallel Tangents Theorem for the ellipse.
The Parabola-Tangent Theorem
If P is a parabola and <L> is a family of lines, then
(1) with one exception, each family contains exactly one tangent to P, and
(2) in the exceptional case, each member of the family cuts C once.
proof
Note that the members of the exceptional family each meet the parabola exactly
once, but are not tangents in any intuitive sense. They do not satisfy clause (2),
which is one reason why we add this clause.
The Hyperbola-Tangent Theorem
If P is a hyperbola and <L> is a family of lines, then there are three cases
(1) the family contains exactly two tangents to P, or
(2) each member of the family cuts H twice, or
(3) one member of the family does not meet H, all others cut H once.
(4) There are exactly two families of the third type.
(5) The tangents to H at P and Q are parallel if and only if PQ is a diameter.
proof
Part (5) is sometimes known as the Parallel Tangents Theorem for the hyperbola.
The special lines in the two families of the third kind are known as the asymptotes
of the hyperbola. Since affine transformations map parallel families to parallel
families and preserve the number of intersections of line and conic, we may
interpret the result as
The Asymptotes Theorem
If H is a hyperbola with asymptotes L and M, and t is an affine transformation
then t(H) has asymptotes t(L) and t(M).
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