Question

what method might be used to detect the existence of the outer planet in this system?
transit method
planet pulsaring method
direct detection
radial velocity method

Explanation:

The transit method is mainly for planets that pass in front of their host - star as seen from Earth. Outer planets may be too far and not transit in a detectable way. The planet pulsaring method is not a standard exoplanet - detection method. Direct detection is difficult for outer planets due to their distance and faintness compared to their host stars. The radial velocity method can detect the gravitational pull of an outer planet on its host star by measuring the star's wobble.

Answer:

radial velocity method