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

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

Leadership · Answer

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.

CareerGeek · Answer

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

exc4libur · Answer

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.

freedom128 · Answer

If a^2 + b^2 = 144 and ab ≠ 0, then -12 NO! b cannot be greater than or equal to 12. (B) 12 and 3 --> YES! Correct answer. b should range as follows: -12

ostrick5465 · Answer

144 = 12^2 a*b eq{0} => b^2 < 12^2 => b<12 => Choice B

eakabuah · Answer

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.

Archit3110 · Answer

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

Posted from my mobile device

lacktutor · Answer

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

unraveled · Answer

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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ScottTargetTestPrep · Answer

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

sachin_27 · Answer

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.

YellowChalk · Answer

why b cant be below 3 ?

Bunuel · Answer

b can be below 3, but that is irrelevant. The question asks for the greatest possible value of b, not all possible values of b.

Username1512 · Answer

I am not sure I understand why a≠12, and b≠12? Any reason for this? I don't see the question specifying this anywhere. Please explain. 
 Bunuel ? Can you please explain?

Bunuel · Answer

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

Notice that we are given that ab ≠ 0.

If a = 12, then a^2 = 144, so b^2 = 0 and b = 0. That violates ab ≠ 0.

Similarly, if b = 12, then a = 0, which also violates ab ≠ 0.

egmat · Answer

Hi Username1512,

Good catch. You're right that the question  never directly says  a≠12 or b≠12. eakabuah's line "ab≠0 means a≠0, b≠0, a≠12, and b≠12" packs two steps into one, and the second step is what's tripping you up.

Here's the split:

-  What ab≠0 says directly:  neither factor can be zero, so  a≠0  and  b≠0 . That's it - nothing about 12 yet.
-  What the equation adds:  a≠12 and b≠12 are not separate given facts - they  follow  from combining ab≠0 with a2 + b2 =  144 .

Watch how b= 12  gets killed. Plug it into the equation:

- If  b = 12 , then b2 =  144 , so a2 = 144 − 144 =  0 , which forces  a = 0 .
- But a = 0 makes ab = 0, and we were told  ab ≠ 0 . Contradiction.

So b can't actually  reach  12  - the only way to hit  12  is to send the other variable to  0 , which is banned. b can get as close as you like (11.99, 11.999...), but never equal  12 . Same logic makes a≠12.

Therefore  the greatest value of b is  just under 12 , landing it in the "between 12 and 3" interval - answer  B .

The key takeaway: the "≠12" isn't stated - it's forced by the equation the moment you refuse to let either variable be zero.

Answer: B

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

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

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