I have been studying conic sections lately and couldnt figure out why we can use the condition D=0 for the quadratic equation having only one root in many of the questions involving finding tangents, which I have read in certain places is called the envelope method. That got me to thinking why the tangent to the conic passes through one and only one point on the conic. A tangent is defined only as a line which just touches the curve at one point. However, it lays no restrictions on whether it could intersect the curve again or not.

So why exactly does a tangent to a circle/ ellipse/ parabola/ hyperbola never cut it again?

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    $\begingroup$ Each branch of a conic section is convex, and a convex curve lies on one side of any tangent, so there cannot be a second intersection point with the same branch. What's left to prove is that a tangent to one branch of a hyperbola cannot intersect the other branch. $\endgroup$
    – dxiv
    Jan 14 at 18:55
  • $\begingroup$ And how exactly can we prove that for a hyperbola ? $\endgroup$ Jan 14 at 19:05
  • $\begingroup$ Geometrically, a tangent to one of the branches is "inside" the common asymptotes on the side of the branch, so it will be "outside" them on the other side. However, both this and the convexity argument are just intuitions, which would need to be formalized for an actual proof. $\endgroup$
    – dxiv
    Jan 14 at 19:25

The "sophisticated" answer is known as "Bezout's theorem". If you have a curve of degree $m$ and a curve of degree $n$, then they intersect in at most $mn$ places. (A "curve of degree $m$" is a curve with its equation given by a polynomial of degree $n$.) So a conic (degree 2) and a line (degree 1) intersect at most $2 \times 1 = 2$ times.

But you have to be careful about how you count the intersections. It turns out that a tangent counts as two intersections. Roughly speaking, this is because if you move the tangent line a little you can make it intersect the conic twice.

The problem here is that you have to prove Bezout's theorem first, which is not exactly the easiest thing in the world.

The more elementary answer is to write out the equations. The conic has equation

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

and the line has equation

$$y = mx + b$$

assuming it's not a vertical line. So you can substitute $y = mx+b$ into the equation for the conic and get

$$Ax^2 + Bx(mx+b) + C(mx+b)^2 + Dx + E(mx+b) + F = 0.$$

Then you get scared that expanding this out will be a pain. But you don't have to! Just observe that $x$ never occurs with power greater than 2. So it's a quadratic in $x$, and has it has at most two solutions.

If the line is a vertical line, then it's $x = k$ for some constant $k$, and substituting this into the conic gives a quadratic in $y$.

  • $\begingroup$ Then how can an assymptote of hyperbola touch the hyperbola at two places, ie at + infinity and - infinity? $\endgroup$ Jan 14 at 19:08
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    $\begingroup$ Well, @MalayKedia , the answer to your question lies in the definitions necessary to state Bézout’s Theorem precisely. Among these is that the whole picture is to lie in projective space, in which we count as one the two “points” at infinity on a line. Continuing further, when you draw the right picture of your situation in the projective plane, you find that the asymptote $x=0$ to the hyperbola $xy=1$ is in fact tangent to the curve at “infinity”. $\endgroup$
    – Lubin
    Jan 14 at 19:56
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    $\begingroup$ That's a good question, @MalayKedia, and Lubin gave a good answer. In practice what it means is that a curve of degree $m$ and a curve of degree $n$ intersect in at most $mn$ points. For example, a line and a conic intersect in at most two points, or two conics intersect in at most four points. The "at most" can be replaced with "exactly" if you take points at infinity and properly take into account multiplicity (for example, the tangent counting twice) $\endgroup$ Jan 14 at 20:31
  • $\begingroup$ So, if I have understood you right, a tangent has a multiplicity of 2, but an asymptote has a multiplicity only of 1, thus it touches curve twice and hence follows bezouts theorem. If I am correct, why are the multiplicity of tangent and asymptote different. Isnt an asymptote just a tangent at infinity? $\endgroup$ Jan 14 at 21:46
  • $\begingroup$ @MalayKedia Sticking on Euclidean space, asymptote never meets the hyperbola. Infinity or infinitesimal never represents a concrete number, they are just tools to find limit. Think back the delta-epsilon approach in calculus. $\endgroup$ Jan 15 at 1:05

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