At what distance would a planet have to orbit the Sun to have its synodic period equal to its sidereal period, assuming circular orbits for Earth and the planet?

Question

Quizplus · Accepted Answer

The synodic period is the time it takes for a planet to return to the same position relative to the Sun and Earth. The sidereal period is the time it takes for a planet to orbit the Sun once. For a planet to have its synodic period equal to its sidereal period, it must be orbiting the Sun at the same distance as Earth. This is because the synodic period is equal to the sidereal period plus the time it takes for Earth to orbit the Sun. If the planet is orbiting the Sun at a different distance than Earth, then its synodic period will be different from its sidereal period.

At What Distance Would a Planet Have to Orbit the Sun