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If a^2 + b^2 = 144 and ab ≠ 0, then the greatest possible value for b

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Bunuel
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If a^2 + b^2 = 144 and ab ≠ 0, then the greatest possible value for b [#permalink] New post   09 Jan 2020, 01:40
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If a^2 + b^2 = 144 and ab ≠ 0, then the greatest possible value for b is between

(A) 16 and 13
(B) 12 and 3
(C) 11 and 4
(D) 10 and 5
(E) 9 and 6
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Re: If a^2 + b^2 = 144 and ab ≠ 0, then the greatest possible value for b [#permalink] New post   09 Jan 2020, 02:08
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If \(a^2 + b^2 = 144 \) and ab ≠ 0, then the greatest possible value for b is between

The sum of squares of 2 integers will not add up to 144 and ab ≠ 0. We can infer that the 'a' and 'b' are not integers.
144 is the square of 12, so, bit less than 12 should be the greatest possible value.
From above statement, we can remove A as the answer just by looking at it and Pick the answer B.
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Re: If a^2 + b^2 = 144 and ab ≠ 0, then the greatest possible value for b [#permalink] New post   09 Jan 2020, 03:44
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If a^2 + b^2 = 144 and ab ≠ 0, then the greatest possible value for b is between

(A) 16 and 13
(B) 12 and 3
(C) 11 and 4
(D) 10 and 5
(E) 9 and 6

\(a^2 + b^2 = 144\)
--> \(b^2 = 144 - a^2\)
--> \(b = \sqrt{144 - a^2}\)

Since, ab ≠ 0, a & b are non zero numbers
So, greatest value of b will ALWAYS be less than 12, which is only implied in option B

IMO Option B
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Re: If a^2 + b^2 = 144 and ab ≠ 0, then the greatest possible value for b [#permalink] New post   09 Jan 2020, 06:25
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If a^2 + b^2 = 144 and ab ≠ 0, then the greatest possible value for b is between

(A) 16 and 13
(B) 12 and 3
(C) 11 and 4
(D) 10 and 5
(E) 9 and 6


a^2 + b^2 = 144, a^2 + b^2 = 12^2
a=0: 0^2+b^2=12^2, b=12; but ab≠0 so, b<12

Ans (B) assuming the range in the answer choices are NOT INCLUSIVE.
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Re: If a^2 + b^2 = 144 and ab ≠ 0, then the greatest possible value for b [#permalink] New post   09 Jan 2020, 06:48
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If a^2 + b^2 = 144 and ab ≠ 0, then -12<b<0 and 0<b<12. b can be an integer or a rational number.

The greatest possible value for b is between:

(A) 16 and 13 --> NO! b cannot be greater than or equal to 12.
(B) 12 and 3 --> YES! Correct answer. b should range as follows: -12<b<0 and 0<b<12

We stop at B because we have found the right answer.

FINAL ANSWER IS (B)

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Re: If a^2 + b^2 = 144 and ab ≠ 0, then the greatest possible value for b [#permalink] New post   09 Jan 2020, 09:57
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\(144 = 12^2\)
\( a*b \neq{0} => b^2 < 12^2\)
\(=> b<12 \)
=> Choice B
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Re: If a^2 + b^2 = 144 and ab ≠ 0, then the greatest possible value for b [#permalink] New post   09 Jan 2020, 20:54
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a^2+b^2=144, where ab≠0.
ab≠0 means that a≠0, b≠0, a≠12, and b≠12. However, a and b can be small fractions.
The word between to modify the range suggests that the limits are not part of the set of numbers considered. The greatest possible value of b is, therefore, between 12 to 3. This is because a can be a small fraction and b can be any number close to 12 but not 12 itself. b can be 11.99 with a being the complementary fraction that makes a^2+b^2=144.

B is the answer.
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Re: If a^2 + b^2 = 144 and ab ≠ 0, then the greatest possible value for b [#permalink] New post   09 Jan 2020, 21:21
144-b^2= + value use plugin technique value of a must be +ve
IMO B 12 and 3




If a^2 + b^2 = 144 and ab ≠ 0, then the greatest possible value for b is between

(A) 16 and 13
(B) 12 and 3
(C) 11 and 4
(D) 10 and 5
(E) 9 and 6

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Re: If a^2 + b^2 = 144 and ab ≠ 0, then the greatest possible value for b [#permalink] New post   09 Jan 2020, 21:53
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If \(a^2 + b^2 = 144\) and ab ≠ 0, then the greatest possible value for b is between -???
--> \(a^2\) -always positive --> \(b^2< 144\)
(b+12)(b-12) <0
-12 <b< 12
Only B satisfies this inequality

The answer is B.
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Re: If a^2 + b^2 = 144 and ab ≠ 0, then the greatest possible value for b [#permalink] New post   09 Jan 2020, 23:21
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If \(a^2 + b^2 = 144\) and ab ≠ 0, then the greatest possible value for b is between

(A) 16 and 13
(B) 12 and 3
(C) 11 and 4
(D) 10 and 5
(E) 9 and 6

\(a^2 + b^2 = 144\)
\((a+b)^2 - 2ab = 144\)
The value of \(a^2\) should be as small as possible so that b takes the maximum value. Hence 2ab is almost negligible.

\((a+b)^2 = 144 + 2ab\)
\((a+b)^2 > 12^2\)
a + b > 12
b > 12 - a (such that b < 12 to satisfy \(a^2 + b^2 = 144\))
b < 12

IMO Answer B.
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Re: If a^2 + b^2 = 144 and ab ≠ 0, then the greatest possible value for b [#permalink] New post   13 Jan 2020, 15:07
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Bunuel wrote:

Competition Mode Question



If a^2 + b^2 = 144 and ab ≠ 0, then the greatest possible value for b is between

(A) 16 and 13
(B) 12 and 3
(C) 11 and 4
(D) 10 and 5
(E) 9 and 6


If a = 0, then the greatest possible value of b is 12. However, since a can’t be 0 (but it still can be very close to 0), the greatest possible value of b can be just slightly less than 12 (for example, if a = 1, then b = √143 ≈ 11.96).

Answer: B
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Re: If a^2 + b^2 = 144 and ab 0, then the greatest possible value for b [#permalink] New post   26 Oct 2023, 08:33
can this be a right approcah?
a^2+b^2-144=0
2ab=144
2*12*6 = 144
therefore highest value is 12 and the other value can be 6 but not below 3.
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Re: If a^2 + b^2 = 144 and ab 0, then the greatest possible value for b [#permalink]
26 Oct 2023, 08:33
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