The plot was that Counselor Troi's mother, Lwazana, who was on board the Enterprise, fell in love with a visiting scientist, Timicin, from Kaelon II. However, their love affair is doomed as Timicin is about to turn 60 and he's getting ready to return to his home planet to die. Lwaxana has difficulty comprehending such custom and asked Picard to intervene. But as the Federation's Prime Directive does not allow the crew to judge or interfere in another's cultural norms and practices, Picard was reluctant to interfere.
Enraged and trying to convince Timicin of how wrong the custom is, at one point, Lwaxana states, "So, you get rid of the problem by getting rid of the people!"
The theme of eutanasia is oft brought up in Star Trek. The Vulcans, too, have a similar practice along with a few other alien cultures.
Other than the resemblance of the Prime Directive to our present day Moral Relativism ("That's just true for you but not for me."), the arguments to justify euthanasia advanced by the various alien cultures in Star Trek, too, resonates with the contentions of contemporary proponents of euthanasia. Many of them claim that those who are suffering from a terminal illness have the "right to die." Two problems arise from this reasoning: 1) how does one define "terminal illness" and where would one draw the line? Cancer? Depression? 2) If to die is a fundamental right, then shouldn't all of us have that right to die, too? Why just those who are terminally ill? This right should not be limited to those who are terminally ill.
I believe that autonomy is not absolute – there are limits to autonomous decisions we make; our autonomy ends where there is tangible evidence of harm to others. This line of rationalization also opens the door to the inflicting of harm where right-to-die can easily slip into duty-to-die just like the people of Kaelon II.
Interesting times we are living in...