Answer:
Given that this is a question with various questions in them, I will try to provide you with the answers for each, and a short explanation.
1. The: astromeric method was used to find a Jupiter-sized planet through careful measurements of the changing position of a star in the sky. This method is the best for finding particularly large planets orbiting away from their stars and its purpose is to detect movements of those bodies against the backdrop provided by other stars. This technique is best used to detect astral bodies close to Earth, big ones and for stars that are close.
2. Discovering planets through the: Doppler method requires obtaining and studying many spectra of the same star. This method is best used for astral bodies that are close by. It detects movements from stars by using something called the Doppler shifts. It can detect massive planets regardless of their orbits, except when the orientation is face on towards Earth.
3. The Kepler mission successfully discovered thousands of extrasolar planets with a spacecraft that searched for transits among some 100.000 stars. Launched in 2009 by NASA, and named after astronomer Johannes Kepler, this enormous telescope was launched to discover planets that were the size of Earth and which were orbiting other stars. The technique it uses to do this is known as the transit method.
4. The transit method is used to find extrasolar planets by carefully monitoring changes in a star´s brightness with time. This method has been the most used, and most successful up to this point for finding the most extrasolar planets. It is the method used by the Kepler mission to discover these extrasolar planets.
