= Solution
Three detection methods are \b[<exoplanet transit photometry>, the <radial-velocity method>, and <exoplanet direct imaging>]. Transits detect obscuration of the star; radial velocities detect stellar reflex motion; imaging separates the planet's own reflected or thermal light from the star.
<Exoplanet transit photometry> supports wavelength-dependent <exoplanet transmission spectra>, while the same orbital geometry supports <exoplanet secondary eclipses> and phase-resolved planetary spectra. <Exoplanet direct imaging> provides resolved light for atmospheric spectroscopy. Stellar radial-velocity discovery alone does not measure an atmosphere, although high-resolution follow-up can separate a moving planetary molecular spectrum through its changing <Doppler effect>.
Two favorable conditions for detecting wide-orbit planets by <exoplanet direct imaging> are:
* \b[Resolvable projected separation:] the angular separation is $\theta\simeq a_\perp/d$, so a wide orbit around a nearby star places the planet beyond the instrument's inner working angle, related to $\lambda/D$ and the <coronagraph> design.
* \b[Sufficient planet-star contrast:] young, massive planets remain intrinsically hot and bright in the <infrared>, improving contrast against the star; starlight suppression and favorable observing wavelengths further help.
A wide orbit improves separation but weakens reflected-light illumination and lengthens the orbital period. These conditions therefore describe imaging sensitivity, not an assertion that every detection method becomes easier at large separation.
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