Could a hidden planet share Earth's orbit?

Speaking to Chris Smith, Francesca Day answered this question for Hannah...

Hannah - Could another planet exist on the other side of the sun that shares our orbit but we never actually see it?

Chris - It's like in the pantomime where the person is always behind the other person it's behind you. Could there be such a planet.

Fran - Probably not. Because even if you didn't see it because the sun was always in the way, we would notice its gravitational effect.

We've measured the orbit of the Earth and other planets very precisely, so we can actually map the influence of the planet on all the other planets. And, if there was an extra one, then even though its gravitational effect is very small compared to that of the Sun, we would still have noticed it...