Find quadratic equation based on 2 tangents

Question

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

Answer 1

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.

Answer 2

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.