Caption: A figure illustrating the Doppler spectroscopy method, one of the two main methods of detecting exoplanets.

A host star and an exoplanet orbit their common center of mass.

The host star is shown at two locations on its orbit.

Actually the figure vastly exaggerates the size of the orbit relative to the star.

The center of mass of planetary system is usually inside the host star---though NOT at the host star's center, of course.

The exoplanet is usually completely undetectable by direct means.

However, the light spectrum of a host star is blueshifted/redshifted when it moves toward/away from Earth.

For an exoplanet, the Doppler shifts of the light spectrum of a host star are minute and result from velocities of order METERS per second executed by the host star.

The detection of exoplanets by Doppler spectroscopy requires refined techniques that have only been developed since circa 1990.

Credit/Permission: © European Southern Observatory (ESO) 2007 (uploaded to Wikipedia by User:ComputerHotline, 2008) / Creative Commons CC BY-SA 3.0.
Image link: Wikipedia: File:The radial velocity method (artist's impression).jpg.
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File: Planetary systems file: doppler_spectroscopy_method.html.