Problem about orbiting of a satelite Problem: There is an satellite that orbits the Earth. Its orbit is circular and its orbital plane is perpendicular to Earth's Equatorial Plane. It orbits is such that a person who is on equator see this satellite on Zenith every 12Hours.

a. Find Satellite's Period.

b. What is distance of satellite to person when it Sets on person's horizon?

My problem with this question is:

For the first part: somewhere said period is 12hours. Somewhere else said the person on equator also moves (Earth Rotation) so 12 hours is P/2 and P= 24hours. What is the right answer?

For the second part: Which of the following figures show the position of satellite when it sets? (Figure 1 or 2?)
I'm treating this as a homework question. You have made a clear attempt, so it is not off topic, but you should clarify the source of the problem.

You will first need to establish if the rotation of the Earth is to be considered (it makes a big difference, but in a homework, sometimes the Earth stops spinning!) You have then correctly found the two possible answers, either 24 or 12 hours.

You can also consider the motion of the Earth around the sun (as the Earth only takes 23hr, 56min to spin once, relative to the stars). What this means is that, after 12 hours, the Earth has turned more than 180 degrees, and there is no orbital plane that passes over the person, and through the origin, and is perpendicular to the equator. However the error is only about 1 degree, so probably can be neglected for the purposes of homework.

For the second part, the first picture is certainly wrong, as the line of sight (in green) passes through the Earth. The second shows the line of sight (now in red) being tangent to the Earth's surface. You will need to use orbital mechanics to find the distance from the Earth's centre to the satellite, and find out the distance from the Earth's centre to its surface (green lines), giving two sides of a right-angled triangle. And the rest is maths.

There are a few details that may affect the answer: you should consider if the motion of the person will affect the answer to part (b) (and if not, why not). You should know that the Earth is not spherical. You should consider whether the optical refraction of light by the Earth's atmosphere should be considered.  However, in a homework problem, such issues can often be ignored, as their effects are small.