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a^2 − b^2 = b^2 − c^2. Is a = |b|?

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a^2 − b^2 = b^2 − c^2. Is a = |b|? [#permalink]

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New post Updated on: 14 Sep 2013, 12:46
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a^2 − b^2 = b^2 − c^2. Is a = |b|?

(1) b = |c|

(2) b = |a|

Hi Please help me in this.I think the answer should be , however answer is

M15-37

Originally posted by nroy347 on 14 Sep 2013, 12:24.
Last edited by Bunuel on 14 Sep 2013, 12:46, edited 2 times in total.
RENAMED THE TOPIC.
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Re: a^2 − b^2 = b^2 − c^2. Is a = |b|? [#permalink]

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New post 14 Sep 2013, 12:53
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a^2 − b^2 = b^2 − c^2. Is a = |b|?

\(a^2 - b^2 = b^2 - c^2\) --> \(a^2+c^2=2b^2\).

(1) b = |c| --> \(b^2=c^2\). From above: \(a^2+b^2=2b^2\) --> \(a^2=b^2\) --> \(|a|=|b|\). Not sufficient.

(2) b = |a|. Not sufficient.

(1)+(2) \(b=|a|\) and \(|a|=|b|\). Still not sufficient. Consider \(a=b=c=1\), for an YES answer and \(a=-1\) and \(b=c=1\), for a NO answer. Not sufficient.

Answer: E.

Or: notice that in the stem and both statements \(a\) is either squared or in the modulus, so both negative and positive values of \(a\) can satisfy each equation. Now, if \(a\) is negative, then the answer would be NO (absolute value of b cannot equal to a negative number) but if \(a\) is not negative, then the answer would be YES.

Hope it's clear.
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a^2 − b^2 = b^2 − c^2. Is a = |b|? [#permalink]

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New post 04 Feb 2015, 21:41
nroy347 wrote:
a^2 − b^2 = b^2 − c^2. Is a = |b|?

(1) b = |c|

(2) b = |a|

Hi Please help me in this.I think the answer should be , however answer is

M15-37


Good question! This is what came to my mind when I read it:

\(a^2 - b^2 = b^2 - c^2\)
The difference between \(a^2\) and \(b^2\) is same as difference between \(b^2\) and \(c^2\) so \(a^2, b^2\) and \(c^2\) form an Arithmetic Progression.

Question: Is a = |b|?

(1) b = |c|
This means \(b^2 = c^2\). So difference between a^2 and b^2 is also 0.
\(a^2 = b^2 = c^2\)
If a = 5 and b = 5, a = |b|
But if a = -5 while b = 5, \(a \neq |b|\)
Not sufficient

(2) b = |a|
Same analysis as above.
If a = 5 and b = 5, a = |b|
But if a = -5 while b = 5, \(a \neq |b|\)
Not sufficient

Using both statements, we see that
If a = 5 and b = 5, a = |b|
But if a = -5 while b = 5, \(a \neq |b|\)
Hence, both together are not sufficient.
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Re: a^2 − b^2 = b^2 − c^2. Is a = |b|? [#permalink]

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Re: a^2 − b^2 = b^2 − c^2. Is a = |b|?   [#permalink] 10 Jun 2018, 01:25
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