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?
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 $y$.