Cleopatra

After the Battle of Pharsalus in 48 BC, where Julius Caesar (Rex Harrison) has defeated Pompey the Great in a brutal civil war for control of the Roman Republic, Pompey flees to Egypt, hoping to enlist the support of the young Pharaoh Ptolemy XIII (Richard O'Sullivan) and his sister Cleopatra (Elizabeth Taylor). Caesar follows him to Egypt, under the pretext of being named the executor of the will of their father, Ptolemy XII. Much to his dismay, Caesar is given Pompey's head as a gift, as the highly manipulated pharaoh was convinced by his chief eunuch Pothinus (Gregoire Aslan) that this act would endear him to Caesar.
As Caesar stays in one of the palaces, a slave named Apollodorus (Cesare Danova) brings him a gift. When the suspicious Caesar unrolls the rug, he finds Cleopatra herself concealed within. He is intrigued with the her beauty and warm personality, and she convinces him to restore her throne from her younger brother.
Soon after, Cleopatra warns Caesar that her brother has surrounded the palace with his soldiers and that he is vastly outnumbered. Counterattacking, he orders the Egyptian fleet burned so he can gain control of the harbor. The fire spreads to the city, destroying the famous Library of Alexandria. Cleopatra angrily confronts Caesar, but he refuses to pull troops away from the fight with Ptolemy's forces to quell the fire. In the middle of their spat, Caesar forcefully kisses her.
The Romans hold, and the armies of Mithridates arrive on Egyptian soil, causing Ptolemy's offensive to collapse. The following day, Caesar is in effective control of the kingdom. He sentences Pothinus to death for arranging an assassination attempt on Cleopatra, and banishes Ptolemy to the eastern desert, where he and his outnumbered army would face certain death against Mithridates. Cleopatra is crowned Queen of Egypt. She begins to develop megalomaniacal dreams of ruling the world with Caesar, who in turn desires to become king of Rome. They marry, and when their son Caesarion is born, Caesar accepts him publicly, which becomes the talk of Rome and the Senate.
After he is made dictator for life, Caesar sends for Cleopatra. She arrives in Rome in a lavish procession and wins the adulation of the Roman people. The Senate grows increasingly discontented amid rumors that Caesar wishes to be made king, which is anathema to the Romans. On the Ides of March in 44 BC, the Senate is preparing to vote on whether to award Caesar additional powers. They assassinate Caesar and split up the empire: Lepidus receives Africa, Octavian Spain and Gaul, while Mark Antony (Richard Burton) will take control of the eastern provinces. However, the rivalry between Octavian and Antony is becoming apparent. Cleopatra is angered after Caesar's will recognizes his adopted son Octavian (Roddy McDowall) instead of Caesarion as his official heir, and angrily returns to Egypt.
While planning a campaign against Parthia in the east, Antony realizes he needs money and supplies, and cannot get enough from anywhere but Egypt. After refusing several times to leave Egypt, Cleopatra gives in and meets him in Tarsus. Antony becomes drunk during a lavish feast. Cleopatra sneaks away, leaving a slave dressed as her, but Antony discovers the trick and confronts the queen. The war is decided at the naval Battle of Actium on September 2, 31 BC where Octavian's fleet, under the command of Agrippa, defeats the Antony-Egyptian fleet. Cleopatra assumes he is dead and orders the Egyptian forces home. Antony follows, leaving his fleet leaderless and soon defeated. Several months later, Cleopatra manages to convince Antony to retake command of his troops and fight Octavian's advancing army. However, Antony's soldiers have lost faith in him and abandon him during the night; Rufio (Martin Landau), the last man loyal to Antony, kills himself. Antony tries to goad Octavian into single combat, but is finally forced to flee into the city.
Octavian and his army march into Alexandria with Caesarion's dead body in a wagon. Octavian has taken Caesarion's ring, which his mother gave him earlier as a parting gift when she sent him to safety.
When Antony returns to the palace, Apollodorus, not believing that Antony is worthy of his queen, convinces him that she is dead, whereupon Antony falls on his own sword. Apollodorus then confesses that he misled Antony and assists him to the tomb where Cleopatra and two servants have taken refuge. Antony dies of his wounds soon after.
Octavian seizes the palace and discovers the dead body of Apollodorus, who had poisoned himself. Octavian receives word that Antony is dead and Cleopatra is holed up in a tomb. Octavian sees her and after some verbal sparring offers her his word that he will spare her, return her possessions, and allow her to rule Egypt as a Roman province in return for her agreeing to accompany him to Rome. She observes Caesarion's ring on Octavian's hand and inquires after her son. When Octavian denies any knowledge of his whereabouts and pledges to allow him to rule Egypt in his mother's absence, Cleopatra knows her son is dead and that Octavian's word is without value. She agrees to Octavian's terms, including a pledge not to harm herself, sworn on the life of her dead son. After Octavian departs, she orders her servants in coded language to assist with her suicide.
She sends her servant Charmion to give Octavian a letter. In the letter she asks to be buried with Antony. Octavian realizes that she is going to kill herself and he and his guards burst into Cleopatra's chamber and find her dressed in gold, and dead, along with her servant Eiras, while an asp crawls along the floor.
Charmion is found kneeling next to the altar on which Cleopatra is lying, and Agrippa asks of her: "Was this well done of your lady?" She replies: "Extremely well. As befitting the last of so many noble rulers." She then falls dead as Agrippa watches.

Question: Where does Caesar banish Ptolemy?
Answer: The eastern desert