Final answer:
The astrometric method, particularly the Doppler-shift method, is best at finding massive planets that orbit very close to their stars since they cause the most significant wobble and radial velocity changes in the star.
Step-by-step explanation:
The astrometric method is used in the search for exoplanets and revolves around detecting the slight motion of a star through space, influenced by the gravitational pull of a planet. When a large planet orbits a star, it can make the star 'wobble' slightly, altering the star's position and/or radial velocity as seen from Earth. This method is particularly sensitive to the gravitational effects of large planets that are close to their stars, also known as 'hot Jupiters', as they induce the most significant wobble and radial velocity changes.
The Doppler-shift method, a specific instance of the broader astrometric technique, involves observing the changes in the radial velocity of a star due to the Doppler effect of its wobbling motion to infer the presence of a planet. Given that planets have to be massive enough to affect the star's radial velocity in a measurable way, the Doppler-shift method is most effective for detecting planets that are both massive and orbit very close to their stars, making option 'a' very close to their stars the correct answer to the question.