Lucia was a longtime smoker, despite the pressure put on her by friends and relatives to stop. when she was recently diagnosed with throat cancer, a few of these friends and relatives expressed very little support or sympathy. this is likely because they viewed lucia's sick role as

The answer would be illegitimate

In the illegitimate sick role, there are stigmas that can be interpreted that the patient causes the disease to their self. In Lucia case, smoking is known to cause cancer and she was already warned by her friends and relatives to stop.
Her friends and relatives would think that she got the disease because of her choice, so she doesn't deserve the support or sympathy.