# All possible tangent lines to a curve through given point

Posted 10 years ago
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 Hi,in Wolfram|Alpha I try something like:tangent line to y=x^2 through (x,y) = (2,2)and expect both possible tangents as result, but I only get one tangent. How can I make W|A show all possible solutions to that type of problem?
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Posted 10 years ago
 Just a little code to show the 2 tangents in question. Have made a slight change to the code to account for negative co-ordinate points, and it will only only account for quadratics, things are very unpredictable if higher powers are used. Clear[x, y, f]; p = {2, 2}; f[x_] := x^2 ;  dif = D[f[x], x]; pts = Solve[f[x] - p[[2]] == dif (x - p[[1]]), x]; pts = x /. pts; t1 = Simplify[f[pts[[1]]]]; t2 = Simplify[f[pts[[2]]]]; m1 = (p[[2]] - t1)/(p[[1]] - pts[[1]]); m2 = (p[[2]] - t2)/( p[[1]] - pts[[2]]);Plot[{m1 (x - pts[[1]]) + t1, m2 (x - pts[[2]]) + t2, f[x]}, {x, pts[[1]] - 5, pts[[2]] + 4}, Epilog -> {PointSize[0.015], Red, Point[{p[[1]], p[[2]]}], Point[{pts[[1]], t1}], Point[{pts[[2]], t2}]}, AspectRatio -> 5/3, ImageSize -> {600, 600}]
Posted 10 years ago
 Hi,There is only one tangent line for a given f(x) at x=x0 if f(x) can be determined uniquely at x0, i.e. d/dx f(x=x0) < |Infinity|Take a look at http://en.wikipedia.org/wiki/TangentI.M.
Posted 10 years ago
 The given point is not part of the curve!  So in this particular case (tangent line to y=x^2 through (x,y) = (2,2)) there are definitly two solutions.In general, there may be zero, one or even multiple solutions to this kind of problem.
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