There is a shooting game with the launcher placed at the origin of a coordinate plane and three circular target disks with radius 1, centered at $(2,2)$, $(4,6)$, and $(8,1)$ respectively. A player selects a positive number $a$ and presses a button. The launcher then fires a laser beam in the direction of point $(1, a)$ (forming a ray). Assume the laser beam can penetrate through the target after hitting it and continue in the original direction (grazing the edge of the disk also counts as a hit). Select the correct options. (1) The laser beam lies on a line passing through the origin with slope $a$ (2) If $a = \frac{3}{2}$, the laser beam will hit the disk centered at $(4,6)$ (3) The player can hit all three disks with just one laser beam (4) The player needs to fire at least three laser beams to hit all three disks (5) If the player fires one laser beam and hits the disk centered at $(8,1)$, then $a \leq \frac{16}{63}$
There is a shooting game with the launcher placed at the origin of a coordinate plane and three circular target disks with radius 1, centered at $(2,2)$, $(4,6)$, and $(8,1)$ respectively. A player selects a positive number $a$ and presses a button. The launcher then fires a laser beam in the direction of point $(1, a)$ (forming a ray). Assume the laser beam can penetrate through the target after hitting it and continue in the original direction (grazing the edge of the disk also counts as a hit). Select the correct options.\\
(1) The laser beam lies on a line passing through the origin with slope $a$\\
(2) If $a = \frac{3}{2}$, the laser beam will hit the disk centered at $(4,6)$\\
(3) The player can hit all three disks with just one laser beam\\
(4) The player needs to fire at least three laser beams to hit all three disks\\
(5) If the player fires one laser beam and hits the disk centered at $(8,1)$, then $a \leq \frac{16}{63}$