If b ≠ 0, does a equal b? (1) a^2 /b^2+4=5 (2) (17a+4b)/7=3a

We need to find, if there is sufficiency to say that a always equal b or a never equals b.

Statement (1) is the equation . Subtracting 4 from both sides of this equation, we have . Multiplying both sides of this equation by b2, we have a^2 = b^2. Now having a^2 = b^2 does not mean that we must have a = b. It is also possible that a = −b. For examples, if a = 4 and b = 4, then a2 = b2, and in this case, a = b, so the answer to the question is "yes." However, if a = 4 and b = −4, then a^2 = a^2, and in this case a is not equal to b, so the answer to the question is "no." Statement (1) is insufficient. We can eliminate choices (A) and (D).

Statement (2) is the equation . Let's try to rewrite this equation. Multiplying both sides of this equation by 7, we have 17a + 4b = 21a. Subtracting 17a from both sides, we have 4b = 4a. Dividing both sides of this equation by 4, we have b = a. Thus, a = b. We can answer the question with a "yes." We have determined that a does equal b. Statement (2) is sufficient. Choice (B) is correct.

Statement 1 : Insufficient
\(\frac{a^2 }{b^2} + 4 = 5\)
\(\frac{a^2 }{b^2} = 1\)
\(a^2 = b^2\)
\(a = b\) or \(a = -b\)
Two different values

Statement 2 : Sufficient
\(\frac{(17a + 4b)}{7} = 3a\)
\(17a + 4b = 21a\)
\( 4b = 4a\)
\( a = b\)

Hence, OA is (B).

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