some special conics

A line has a linear equation. A conic has a quadratic equation. It follows that
a line and a conic meet at most twice. If the equations result in two coincident
roots, then we say that the line is a tangent to the conic at the point where
they meet.

For the general conic, obtaining tangents algebraically is tedious. We shall look
at some special cases.

The Parabola P : y2 = x.

(1) The line y = k cuts P exactly once.
(2) The line x = ny+c

  • cuts P twice if n2+4c > 0,
  • is a tangent to P if n2+4c = 0, and
  • does not meet P if n2+4c < 0.

Note that each family {L : y = nx+c, c real} of parallel lines contains
exactly one tangent, namely that with c = n2/4.

Although we are used to the hyperbola as x2-y2=1, we can rotate through the
angle π/4 anticlockwise to get the equation xy = 1, with asymptotes x = 0 and
y = 0.

The Hyperbola H : xy = 1.

(1) The lines x = 0 and y = 0 do not meet H.
(2) For c ≠ 0, the lines x = c and y = c each meet H exactly once,
(3) For m ≠ 0, the line y = mx+c,

  • cuts H twice if m > 0 or c2+4m > 0,
  • is a tangent to H if c2+4m = 0,
  • does not meet H if c2+4m < 0.


Note that the family {L : y = mx+c, c real} contains

  • no tangents if m > 0, and
  • two tangents if m < 0, these are given by the c with c2=-4m.
The families {L : x = c, c real} and {L : y = c, c real} are rather different.
Each contains one line which does not meet H - these are the asymptotes -
while the other lines cut H exactly once, but are not tangents.

Although we do not regard the circle as a euclidean conic, the results for the
circle give a useful pointer to the situation for the ellipse. Here the proof is
purely geometrical.

The Circle

For any circle C, each family of parallel lines contains exactly two tangents
to C, and these occur at diametrically opposite points of C.


This generalises to ellipses, as we shall see when we look at affine geometry.

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