Find the general equation of a parabola given a tangent line and point of intersection

CactUs

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Sep 14, 2020
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Find the equation of the parabola tangent to ax + b at x=c where a, b, c are constants

So I know that I'm solving for a parabola Px2 + Qx +R where:

f(c) = Pc2 +Qc + R = ac + b
f'(c) =Pc + Q = a

And I know that since there is more than one solution parabola for each a, b, and c, I'll have an expression with x, y, a, b, c, and one other variable at the end, but I just don't see how this system of equations can possibly be solved for what I want. Please help!

Thanks
 

Subhotosh Khan

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Find the equation of the parabola tangent to ax + b at x=c where a, b, c are constants

So I know that I'm solving for a parabola Px2 + Qx +R where:

f(c) = Pc2 +Qc + R = ac + b
f'(c) =Pc + Q = a

And I know that since there is more than one solution parabola for each a, b, and c, I'll have an expression with x, y, a, b, c, and one other variable at the end, but I just don't see how this system of equations can possibly be solved for what I want. Please help!

Thanks
You CANNOT define a unique parabola to be tangent to a given line. There will be \(\displaystyle (\infty )^2\) possible parabolas.
 

CactUs

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Sep 14, 2020
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But is it possible to find the general form of the equation of such a parabola? That is, can the constants a, b, and c in ax2 + bx +c be represented in terms of the equation of the tangent line and the x-value of the point of intersection? And is there any simple proof for why there are infinite parabolas?

Thank you all so much for your time
 

Dr.Peterson

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Nov 12, 2017
Messages
10,191
Find the equation of the parabola tangent to ax + b at x=c where a, b, c are constants

So I know that I'm solving for a parabola Px2 + Qx +R where:

f(c) = Pc2 +Qc + R = ac + b
f'(c) =Pc + Q = a

And I know that since there is more than one solution parabola for each a, b, and c, I'll have an expression with x, y, a, b, c, and one other variable at the end, but I just don't see how this system of equations can possibly be solved for what I want. Please help!
First, correct your derivative. You dropped a 2.

Then you will have two equations in three unknowns, P, Q, and R. You can eliminate one of these; I chose P, but I'm not sure it's the best choice. Then you will have an equation in the other two unknowns, so you can solve for one of them, e.g. Q, in terms of the other. Then you can obtain expressions for, say, P and Q in terms of a parameter R.

This proves that there are infinitely many such parabolas; but it is obvious that there are infinitely many parabolas through a given point with a given slope, just by trying to sketch some. Since you said yourself that "there is more than one solution parabola", I think you recognize this.

I haven't put any time into it, but there are probably nicer ways to do this in terms of some different parameters. One thought I have is that you could specify that the parabola also passes through the point (t, ac+b); that is, specify the other place where y is the same as the given point. Then t might be a convenient parameter. Or, you could use the x-coordinate of the axis of symmetry as a parameter. Then you would be trying to express P, Q, and R in terms of your parameter.
 
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