Imaginary vs. Imaginative

Reading that a child in Texas was suspended for bringing “an imaginary ring” to school, I marveled that the school officials were able to detect the ring’s presence.

Here’s the headline:

Texas School Suspends 9-Year-Old for Terrorism
Because He Brought Imaginary Hobbit Ring To School

In fact, the child brought a real ring to school, presumably a replica of the ring carried by Bilbo Baggins in the Peter Jackson movie The Hobbit.
The ring was real, but its magical powers were imaginary.

The English word image derives from Latin imago. One meaning of image is “mental picture.” Something imaginary or imagined exists in the mind.

Here is a review of image words with definitions and examples:

imagination (noun): The power or capacity to form internal images or ideas of objects and situations not actually present to the senses. Example: It is because of the development of the imagination during childhood that adults are able to do many of the tasks that daily life demands. 

imaginary (adjective): Existing only in imagination or fancy; having no real existence; not real or actual. Example: Lilliput is an imaginary country visited by Gulliver.

imaginative (adjective): relating to, or concerned in the exercise of imagination as a mental faculty. Example: Imaginative Artists Find New Ways to Deal With the Western Landscape Tradition

imagine (verb): conceive in the mind. Example: The universe is not only stranger than we imagine, it is stranger than we can imagine.

imagined (past participle): invented, created in the imagination. Example: The second basic axiom concerning power is that the powerful always try to create an outside enemy, real or imagined, to bind the followers to the leaders. 
Errors also occur with the pairs imaginary/imagined and imaginary/imaginative:

Incorrect: It is easy to perceive a country as an imaginary enemy.  
Correct : It is easy to perceive a country as an imagined enemy.
The country actually exists, so it can’t be imaginary. It can, however, be “an imagined enemy.”

Incorrect: Children learn from experience: from what happens around them, from what they see, hear, smell, taste and touch. To absorb those experiences and make sense of the world, they need to be engaged in imaginary play.  
Correct : Children learn from experience: from what happens around them, from what they see, hear, smell, taste and touch. To absorb those experiences and make sense of the world, they need to be engaged in imaginative play.

The play is not imaginary; it is real. Because the child is exercising imagination, the play is imaginative.

Source: http://www.dailywritingtips.com/imaginary-vs-imaginative/