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I would like to know a way to find an quadratic equation that had 2 given tangents:

For example here is 2 tangents equations:

  • y = 1/2 x
  • y = 2 x + 2

and 2 abscisses

  • x = 0
  • x = 3

Is there a simple way to find a quadratic equation that saitisfies those two tangents at respective abscisses?

I'm working in Clojure if it needs to be computer related

Sorry for my bad english

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    $\begingroup$ This question appears to be off-topic because it is about mathematics. Questions on Stackoverflow should be at least tangentially related to computer programming. $\endgroup$
    – Joni
    Jan 2, 2014 at 10:43
  • $\begingroup$ I have flagged it for moderator attention. $\endgroup$
    – Ali Gajani
    Jan 2, 2014 at 10:44
  • $\begingroup$ You should be a bit more clear: exactly what do you mean by 'quadratic equation', and what do you mean by an equation 'satisfying a tangent'? $\endgroup$
    – Andreas Rejbrand
    Jan 2, 2014 at 10:49
  • $\begingroup$ In comments below you say you are looking for a parabola, and not just any quadratic equation. Do you have any limitations or preferences on the kind of a parabola, for example if its axis should be parallel to the X or Y axis? $\endgroup$
    – Joni
    Jan 2, 2014 at 18:29

2 Answers 2

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Given that:

y = 1/2 x         <=>    1/2 x - y = 0
y = 2 x + 2       <=>    2 x + 2  - y = 0

Therefore

(1/2 x - y)(2 x + 2  - y) = 0

is one quadratic equation that has these two lines as tangents. In fact, if you plot its solutions you'll find they consist of the two lines.

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  • $\begingroup$ Yes, I realised the wording was unclear. $\endgroup$
    – Joni
    Jan 2, 2014 at 10:44
  • $\begingroup$ Although your answer is technically correct, I suspect the OP is actually looking for parabolae! :) $\endgroup$
    – Andreas Rejbrand
    Jan 2, 2014 at 10:46
  • $\begingroup$ yes i'm searching for a parabolae, maybe i can specify my question by giving the abscisses at wich the parabolae is tangent to those linear equations $\endgroup$
    – szymanowski
    Jan 2, 2014 at 11:02
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From the little that you provide, I infer that you are looking for a parabola equation of the form

y = A x^2 + B x + C

tangent to the two lines at the given abscissas.

The slope is given by

y' = 2 A x + b

Expressing the equal slope conditions, we have

2 A 0 + B = 1/2
2 A 3 + B = 2

giving A = 1/4, B = 1/2.

Expressing the equal ordinate conditions, we have

A 0^2 + B 0 + C = 0 / 2
A 3^2 + B 3 + C = 2 3 + 2

leading to 9/4 + 3/2 = 8 !!!

There is no solution, as there are too many constraints.

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