Capital letter to be used: Bombay
Article to be corrected: a

Explanation:

Capital letter to be used:
In this sentence, the word "Bombay" is a proper noun, as it refers to a specific place. Proper nouns are always capitalized, so the correct usage here would be to capitalize the first letter of "Bombay".

Article to be corrected:
The indefinite article "a" should be changed to the definite article "the" in this sentence. The word "the" is used to refer to a specific noun or noun phrase that is already known to the speaker and the listener. Since the sentence is referring to "the miracles of modern science", it suggests that the miracles being mentioned are already known or have been previously mentioned. Therefore, the correct article to use in this context is "the".

Revised sentence:
They had come with high hopes in the miracles of modern science to Bombay.

Summary:
In summary, the correction to be made in the given sentence is to capitalize the first letter of "Bombay" and change the indefinite article "a" to the definite article "the". The revised sentence would be: "They had come with high hopes in the miracles of modern science to Bombay."