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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.
The question (x - a)^2 = (x - b)^2 is equivalent to x = (a + b)/2 for the following reason.
(x - a)^2 = (x - b)^2
⇔ (x - a)^2 - (x - b)^2 = 0
⇔ ((x - a) - (x - b)) ((x - a) + (x - b)) = 0
⇔ (-a + b)(2x - (a + b)) = 0
⇔ (2x - (a + b)) = 0 (by dividing both sides by -a + b since a ≠ b)
⇔ 2x = (a + b) (by adding ( a + b) to both sides)
⇔ x = (a + b)/2 (by dividing both sides by 2)
So, we have to find the value of a + b.
Thus, look at condition (2). It tells us that a + b = 7, which is exactly what we are looking for. The answer is unique, and the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Condition (1) tells us that a – b = 3, from which we cannot determine the unique value of a + b. The answer is not unique, and the condition is not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Condition (2) ALONE is sufficient.
Therefore, B is the answer.
Answer: B
