Tenchi's family life is a bit strange. He lives with his father ( who is a pervert ) when school
is in session. And as was stated before he spends the summer with his grandfather who was his mother's father. Unfortunately
for Tenchi, his mom died when he was very young ( Much of the circumstances surrounding
Tenchi and his mom are revealed in Tenchi Muyo in Love ).

Throughout his adventures Tenchi learns more about his potential to be the next
ruler of the Jurai, and about his uncanny ability to make every girl around him
fall in love with him ( Ayeka and Ryoko being the 2 main contenders ).

Tenchi-ken is the name of the sword Yosho used to defeat Ryoko 700 years ago.
At the beginning of the first OAV series, it contains all three of
Ryoko's Power Gems embedded in its hilt,
effectively sealing Ryoko powerless in the cave.

When Tenchi first finds it, Tenchi-ken's blade is rusted and shatters as Tenchi mutters something to the effect of "The sword that can cut through stone ?!?" ( sarcastically )... And of course after it shatters because the "fake" blade was beyond completely rusted , the stone which was a "lock" on Ryoko's prison slices in half ; )

Later in the first episode, a blue energy blade similar to
Ryoko's Energy Sword is emitted from
its hilt. Only direct members of the Jurai bloodline can wield this sword.

When introduced to Ayeka and Sasami in episode 2, we are told that
Tenchi-ken is Yosho's key to controlling the Juraian Trees. Sasami and Ayeka also have their own keys but since Yosho was the Prince of Jurai, his key works on all Juraian trees.