For what values $a, b$ parabola $y = ax^2$ will be tangent with the line $y=2x+b$?
Do I need the derivative of the equations? Or one of them then compare between them?
- $y=ax^2$
- $y=2x+b$
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Michael Albanese
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Ofir Attia
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If they are tangent, that means that their intersection is only one point – Koam Apr 17 '13 at 12:30
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It might be helpful to the OP to post more details about this statement to provide more guidance as an answer. Regards – Amzoti Apr 17 '13 at 12:50
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It is a little dangerous to characterize tangent lines as "intersecting the curve in one point" in general, since the tangent line may intersect in more than one point. (E.g. the line tangent to $\sin(x)$ at $x=\pi/2$. (Granted, though, that it is harmless in the case of a parabola :) ) – rschwieb Apr 17 '13 at 13:34
3 Answers
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Hints:
1-Two functions intersects, so we have to solve $ax^2=2x+b$.
2- The derivation of $y=ax^2$ (which is $2ax$) is $2$ when $x$ is one of above quadratic equation's solutions.
Mikasa
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1@BabakS.: I think different problems call for different answers and sometimes clear hints are spot on! Regards – Amzoti Apr 19 '13 at 05:13
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Hint: If they are tangent, that means that their intersection is only one point. That means that: $$ ax^2 = 2x+b $$ has only one solution!
guaraqe
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tangent to the parabola will be of the form: $$y^1 = 2ax$$ This is equal to $2x+b$ only when: $$2ax = 2x+b$$ Comparing co-efficients gives you: $$(a,b)=(1,0)$$
lsp
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