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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.

We can put BC = BF = x.
Put BC = a = q + x, AC = b = p + q and AB = c = p + x.
Then the area of triangle ABC = (1/2)ac and x = (a-b+c)/2, p = (b-(a-c))/2 and q = (b+(a-c))/2
Then we have
pq = {(b-(a-c))/2}{(b+(a-c))/2}
pq = {b^2 + b(a-c) – b(a-c) - (a-c)2}/4
pq = (b^2 – (a-c)2)/4
pq = {b^2 – (a-c)(a-c)}/4
pq = {b^2 – (a2 – ac – ac + c2)}/4
pq = (b^2 - a2 - c2 + 2ac)/4
pq = ac/2.
Thus, the area of the triangle ABC is pq, and condition 1) is sufficient.

Condition 2)

Since there are a lot of possibilities for p and q, condition 2) does not yield a unique solution, and it is not sufficient.

Therefore, A is the answer.
Answer: A