Prove this variation: If a+b ∝ a-b, prove that a^2+b^2 ∝ ab

AI Thread Summary
The discussion centers on proving that if a + b is proportional to a - b, then a² + b² is proportional to ab. The initial derivation shows that a + b = k₁(a - b) leads to a² + b² being expressed in terms of ab with a new constant k₂. However, several participants argue that the proof is flawed, citing counterexamples and questioning the assumptions about the constants. They emphasize that proportionality requires independent constants and that the problem lacks clarity regarding the nature of a and b. Ultimately, the consensus is that the proof does not hold under the given conditions.
RChristenk
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Homework Statement
If ##a+b \varpropto a-b ##, prove that ##a^2+b^2 \varpropto ab##
Relevant Equations
Basic fractions and algebra
##a+b \varpropto a-b## means ##a+b = k_1(a-b)##

##(a+b)^2=k_1^2(a-b)^2##

##a^2+b^2+2ab=a^2k_1^2+b^2k_1^2-2abk_1^2##

##2ab(1+k_1^2)=a^2(k_1^2-1)+b^2(k_1^2-1)##

##2ab(k_1^2+1)=(a^2+b^2)(k_1^2-1)##

##a^2+b^2=ab(\dfrac{2k_1^2+2}{k_1^2-1})##

Since ##a^2+b^2 \varpropto ab## means ##a^2+b^2=abk_2##, compared to what I got, I can say ##k_2 = \dfrac{2k_1^2+2}{k_1^2-1}##. Therefore ##a^2+b^2 \varpropto ab## has been proven. Is this a correct? Thanks.
 
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##a = 3, b = 1 \ \Rightarrow \ (a+b) = 2(a -b)##

##a^2 + b^2 = 10, \ ab = 3##
 
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Have you stated the problem exactly and completely? Maybe I'm missing something but it seems false to me.
 
FactChecker said:
Have you stated the problem exactly and completely? Maybe I'm missing something but it seems false to me.
It's definitely false, given I provided a counterexample!
 
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RChristenk said:
Homework Statement: If ##a+b \varpropto a-b ##, prove that ##a^2+b^2 \varpropto ab##
Relevant Equations: Basic fractions and algebra

##a+b \varpropto a-b## means ##a+b = k_1(a-b)##
What seems to be missing in the problem statement here is the idea that ##a^2 + b^2## must be shown to be in the same proportion to ab as a + b is to a - b.

IOW, if a + b = k(a - b), then ##a^2 + b^2 = kab##.
 
Mark44 said:
What seems to be missing in the problem statement here is the idea that ##a^2 + b^2## must be shown to be in the same proportion to ab as a + b is to a - b.

IOW, if a + b = k(a - b), then ##a^2 + b^2 = kab##.
That doesn't seem possible.
 
PeroK said:
That doesn't seem possible.
I agree. My comment was about I believe what the intent of the problem was, taking into account your counterexample. As a very simple example of what I am talking about, show that if ##x \varpropto y## then ##2x \varpropto 2y##. In the two resulting equations, the same constant applies to both.

What the OP is doing is something like this:
Show that ##x \varpropto y \Rightarrow x^2 \varpropto y^2##.
##x \varpropto y \Rightarrow x = ky## for some nonzero k
##x = ky \Rightarrow x^2 = k^2y^2 \Rightarrow x^2 = K y^2 \Rightarrow x^2 \varpropto y^2##

But if x = 4 and y = 2, then x = 2y, but ##x^2 = 16 \ne 8 = 2y^2##.
x and y aren't in the same proportion as ##x^2## and ##y^2##.
 
The question just states if ##a+b \varpropto a-b##, prove that ##a^2+b^2 \varpropto ab##.

I think the constants are different. So ## a+b = k_1(a-b)## and ##a^2+b^2 = k_2ab##.

##\Rightarrow (a+b)^2=k_1^2(a-b)^2##

##\Rightarrow a^2+2ab+b^2=a^2k_1^2-2abk_1^2+b^2k_1^2##

Now using ##a^2+b^2 = k_2ab## and replacing the left side of the above equation, get:

##\Rightarrow k_2ab+2ab=a^2k_1^2-2abk_1^2+b^2k_1^2##

##\Rightarrow k_2ab+2ab+2abk_1^2=(a^2+b^2)k_1^2##

##\Rightarrow ab(k_2+2+2k_1^2)=(a^2+b^2)k_1^2##

##\Rightarrow a^2+b^2 = ab\dfrac{(2k_1^2+k_2+2)}{k_1^2}##

So since ##\dfrac{(2k_1^2+k_2+2)}{k_1^2}## is a new constant of variation, ##a^2+b^2 \varpropto ab## is proven. That's just what I'm thinking.
 
RChristenk said:
The question just states if ##a+b \varpropto a-b##, prove that ##a^2+b^2 \varpropto ab##.

I think the constants are different. So ## a+b = k_1(a-b)## and ##a^2+b^2 = k_2ab##.

##\Rightarrow (a+b)^2=k_1^2(a-b)^2##

##\Rightarrow a^2+2ab+b^2=a^2k_1^2-2abk_1^2+b^2k_1^2##

Now using ##a^2+b^2 = k_2ab## and replacing the left side of the above equation, get:

##\Rightarrow k_2ab+2ab=a^2k_1^2-2abk_1^2+b^2k_1^2##

##\Rightarrow k_2ab+2ab+2abk_1^2=(a^2+b^2)k_1^2##

##\Rightarrow ab(k_2+2+2k_1^2)=(a^2+b^2)k_1^2##

##\Rightarrow a^2+b^2 = ab\dfrac{(2k_1^2+k_2+2)}{k_1^2}##

So since ##\dfrac{(2k_1^2+k_2+2)}{k_1^2}## is a new constant of variation, ##a^2+b^2 \varpropto ab## is proven. That's just what I'm thinking.
But this definition of "proportionality" is tautological. Any two quantities ##x## and ##y## are "proportional" (##y=kx##) if one is free to set ##k=y/x##. A proper proportionality requires that ##k## be independent of ##x,y##.
 
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  • #10
RChristenk said:
The question just states if ##a+b \varpropto a-b##, prove that ##a^2+b^2 \varpropto ab##.

I think the constants are different. So ## a+b = k_1(a-b)## and ##a^2+b^2 = k_2ab##.

##\Rightarrow (a+b)^2=k_1^2(a-b)^2##

##\Rightarrow a^2+2ab+b^2=a^2k_1^2-2abk_1^2+b^2k_1^2##

Now using ##a^2+b^2 = k_2ab## and replacing the left side of the above equation, get:

##\Rightarrow k_2ab+2ab=a^2k_1^2-2abk_1^2+b^2k_1^2##

##\Rightarrow k_2ab+2ab+2abk_1^2=(a^2+b^2)k_1^2##

##\Rightarrow ab(k_2+2+2k_1^2)=(a^2+b^2)k_1^2##

##\Rightarrow a^2+b^2 = ab\dfrac{(2k_1^2+k_2+2)}{k_1^2}##

So since ##\dfrac{(2k_1^2+k_2+2)}{k_1^2}## is a new constant of variation, ##a^2+b^2 \varpropto ab## is proven. That's just what I'm thinking.
Apart from when zero is involved, all real numbers are proportional to each other. This question only makes sense if the numbers involved are integers. In this case, if ##a, b \ne zero##, then:
$$a^a + b^2 = (\frac{a^2 + b^2}{ab})ab$$And there is no need to do any calculations. And nothing to prove.

Moreover, ##a + b = k(a - b)## is true for all real numbers ##a, b##. So, there is no "if" involved. It's only when ##a, b## and ##k## are integers that this condition makes sense.
 
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  • #11
RChristenk said:
Homework Statement: If ##a+b \varpropto a-b ##, prove that ##a^2+b^2 \varpropto ab##
Perhaps ##a## and ##b## are supposed to be variables or parameters? Then ##a+b \varpropto a-b ## implies ##a \varpropto b ## and the result is trivial.

I originally assumed it was an integer equation. The question needs some context.
 
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  • #12
PeroK said:
Perhaps ##a## and ##b## are supposed to be variables or parameters? Then ##a+b \varpropto a-b ## implies ##a \varpropto b ## and the result is trivial.
IMO, this is the answer that the problem had in mind.
 
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