Question

which of these is the best location to place a telescope

Answer 1

This is a very good question, but actually the question needs to be rephrased. What are we wanted to view in the telescope?

If I want to see the girl across the street undress then I want the telescope in my house and I want it pointing at her house. LOL

All joking aside.

Typically when you are looking into a telescope you are viewing something astronomical. Mountains are a good place to put a telescope because usually you avoid light pollution and when you are at a higher altitude atmospheric turbulence is lessened. Turbulence in the atmosphere makes the stars twinkle.

Basically the reason telescopes are built on tall mountains is to get away from the distortion of starlight due to the atmosphere

A good place to have a telescope is on a mountain, the BEST place is to put the telescope in space like in an orbit around the Earth like the Hubble Space Telescope or on the Moon. This way the Earth's atmosphere is completely avoided.

Answer 2

Funny and helpful! Some other points to think about:
Positioning the telescope near-ish to the equator helps in keeping year-round temperature variations down, but that is a minor point. One telescope cannot see all the stars in the sky, only half of it. If you want to view northern hemisphere objects, you must biuld in the northern hemisphere! (and vice versa).
Next, astronomers (well, the engineers) also want a smooth air flow, to also reduce atmospheric blurring from turbulence. As such sites near to a coast help this, together with smooth land formations.
In terms of space, that does get you the best resolution, but there are still drawbacks.. 1st is faint source detection. This requires the biggest mirror possible, which are built on Earth (diameter ~10m currently), as launching things this big is difficult/impossible. This links to the 2nd problem of cost, space telescopes cost more. Final drawback of space is the difficulty in repairs..