Not coming ON

If it is a constant low-pitched buzzing it is probably mains hum which can be difficult to eliminate.
There are two sorts of mains hum; conducted and radiated. Mains operated audio equipment suffers from this when the power supply smoothing is ineffective or when a hum loop has been inadvertently created. High gain preamplifiers with non-shorting inputs, high impedance peripherals and long leads make equipment more vulnerable.
Even battery powered equipment with a high gain can suffer from mains hum received like low frequency radio waves radiated from nearby mains operated equipment and supply wiring.

check AC - DC adapter could be faulty
Check surge protection if plugged into a surge protected powerboard
Could have a crook powerboard

When your laptop is functionning and plugged, does the battery indicates that is is recharging?

As stated in the manual:
The green power-ON indicator lamp lights up when the tool is plugged to the mains. If the indicator lamp is lit but the tool does not start even if the tool is switched ON, the carbon brushes may be worn out, or the motor or the switch may be defective. If the indicator lamp does not light up, the mains cord or the indicator lamp may be defective.

From the owner's manual: Indicator lamp The green power-ON indicator lamp lights up when the tool is switched ON. If the indicator lamp is lit but the tool does not start, the carbon brushes may be worn out, or the electric circuit or the motor may be defective. If the indicator lamp does not light up and the tool does not start, the ON/OFF switch or the mains cord may be defective.

The easiest to check is the brushes. Most newer tools have brushes with a spring loaded pin in them so when they wear down the pin pops and stops the motor so you don't wear the armature. Even without the pin the wire from the contact to the brush is short so the spring doesn't contact and ruin the armature. If the brushes are OK, you'll have to check the switch for continuity.

Do you have any LEDs indicating that the projector is getting power?
If not, it is unlikely to be the lamp which is the problem and more likely a problem with the mains supply outlet, extension cable, mains input cable, mains input fuse (all easy to check) or maybe the internal power supply (requiring professional repair).
If there are LEDs lit they will indicate a fault, for which you will either need to post the exact sequence and timings, or consult the manual.

Can you describe the symbols that are next to the lights that are still lit?
If your laptop is still attached to mains power (ie your power pack that feeds DC voltage to your laptop) the battery pack may be still charging (indicated by a light) The battery pack is internal to your laptop and this powers your laptop when the power pack is not connected to mains.
Try disconnecting your mains power and see if a light goes off. If it does then that light is the light which indicates charging is in progress.
Other lights that are typical to laptops are:
a. hard drive activity (usually lights when the hard drive is being used) Usually has an adjacent symbol which looks like a metal can or cylinder.
b. bluetooth (wireless connection to other bluetooth devices such as cell phones, other bluetooth devices such as another computer) Adjacent symbol looks like two triangles side by side.
c. sleep (usually lit when your laptop goes to sleep and saving power) This may be covered by the power on lamp mentioned below, and the power light may flash to indicate the laptop is in sleep mode to save battery power.
d. power on (lights when your laptop is ON)
e. battery charging (as described above). There is often a symbol adjacent to this light that looks like a small battery.
Good luck,
Steve (Melb, Australia)

needs a new main board.

friend

the problem clearly shows the fault is in the motherboard

thanks

It'll likely be right behind where the power cord from the TV comes from.