following on what b^3 and argonaut said, there is a general method of finding tangential points, I'm in a proofy mood these days so...
Useless computational technique for methods? Yes
Useful exercise? Yes
oh btw if you're a methods student, then just ignore what's beneath, it's just my random musing, as you can see i'm bored and what do you do when you're bored? YOU START PROOFING
Consider any curve
)
, tangent is given by
and assume we need to find the tangent at point
)
Obviously,
)
and let
)
be equation of the tangent line.
Note that
Then
)
:
(x-X)+\frac{\partial f}{\partial y}(X, Y)(y-Y) = 0)
Assume our curve f(x) is an algebraic curve, then note that homogeneity exists if
 = \lambda^k f(x,y,z))
, note the general case
 = \phi^kf(x))
and so applying Euler's Theorem yields the generality
 = k f(\mathbf{x}))
, then:
x +\frac{\partial h}{\partial y}(X,Y,Z)y+\frac{\partial h}{\partial z}(X,Y,Z)z=0)
Applying this to our algebraic curve, yields the homogeneous equation:
Applying this above equation yields our tangential line:
x+\frac{\partial f}{\partial y}(X,Y)y+u_{n-1} +\cdots + nu_0=0)
(clearly z=1)
this is pretty useful especially when finding tangents to homogeneous polynomials (
http://en.wikipedia.org/wiki/Homogeneous_polynomial) and a few other families
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