1)I saw the boy with the help of a binocular. 2)The boy was using the binocular and I saw him. Please tell the correct answer with the reason.

As explained in the answer above, the two sentences given in the question have distinctly different meanings. Both the sentences are grammatically correct. However, to improve the understanding of how one sentence can have ambiguous meanings, and how we can improve the clarity of sentences. I am giving below a grammatically correct sentence, which could be interpreted to mean any of the two statements made in the question. We will then see how this sentence can be modified to remove the ambiguity in its interpretation.

Consider the following sentence:

I saw the boy using a binocular.

Here the phrase "using a binocular" may mean that I was able to see because I was using a binocular. Alternately, the phrase could describe that when I saw the boy, he was using a binocular. The ambiguity of the sentence can be removed by modifying the sentence as:

I saw the boy by using a binocular.

Or

I saw the boy who was using a binocular.

The first of the two modifies sentence has the same meaning as sentence 1) of the question, and the second  modified sentence has the same meaning as sentence 2) of the question.