The question asks to identify the sentence where the underlined word is used correctly, focusing on the distinction between similar words like 'elude', 'elusive', 'allude', 'allusive', and 'illusory'.

Option A: Incorrect Usage

Analysis: The sentence states, "When the teacher eludes to different authors, he is being elusive."

• 'Eludes' means to escape or avoid something.
• 'Alludes' means to suggest or hint at something indirectly.
• The correct verb here should be 'alludes', not 'eludes'. A teacher hints at authors, they don't escape from them.
• Furthermore, 'elusive' means difficult to catch or understand. While a teacher's point might be hard to grasp, the primary error is the misuse of 'eludes'.

Therefore, option A is incorrect.

Option B: Correct Usage

Analysis: The sentence states, "When the thief keeps eluding the police, he is being elusive."

• 'Eluding' means successfully escaping or avoiding the police. This fits the context perfectly.
• 'Elusive' describes something or someone that is difficult to catch or find. A thief actively eluding the police is, by definition, elusive.

Both words are used correctly in this context. Therefore, option B is the correct answer.

Option C: Incorrect Usage

Analysis: The sentence states, "Matters that are difficult to understand, identify or remember are allusive."

• Matters that are difficult to understand, identify, or remember are best described as 'elusive'.
• 'Allusive' means characterized by indirect reference or suggestion.

The sentence incorrectly uses 'allusive' when 'elusive' is the appropriate term. Therefore, option C is incorrect.

Option D: Incorrect Usage

Analysis: The sentence states, "Mirages can be allusive, but a better way to express them is illusory."

• 'Mirages' are deceptive appearances or optical phenomena that are not real.
• 'Illusory' means based on illusion; not real. This word accurately describes mirages.
• 'Allusive' means characterized by indirect reference. This word does not describe a mirage. Mirages are directly deceptive, not indirectly suggestive.

While 'illusory' is correct for mirages, 'allusive' is incorrectly applied. Therefore, option D is incorrect.

Conclusion

Option B is the only sentence where the underlined words ('eluding' and 'elusive') are used correctly according to their definitions.