This is a labelled toy model, not a real telescope. One honest idea runs everything: you can find something you cannot see by watching what it pulls. The star and its hidden planet both circle a shared balance point, so the star draws a tiny circle of its own. The geometry here is exact - the balance point never moves, the two partners stay on opposite sides, and the circle repeats exactly on time. You can audit it tick by tick.
What is real: Neptune was found with math before anyone saw it - its pull was bending Uranus's path, so astronomers predicted where to look. Today, most planets around other stars are known only by their effects: the star's tiny wobble, or the star's light dimming as a planet crosses. NASA's own pages say all of this. The wobble size really does grow with the hidden planet's heft, and finding the pattern really does take many patient measurements.
Left out on purpose: the wobble is drawn enormously bigger than life (a real star's circle is far too small to see by eye), the tug-of-light trick (astronomers really measure stretched and squeezed starlight), planets that pass in front of their star, and systems with many planets, whose wobbles mix together. No real sky locations or personal details are used; nothing about you leaves this page.
Sources: NASA Science ("How We Find and Characterize" exoplanets - "Radial Velocity: Reading the Wobble") and NASA Science's Neptune overview ("the first planet located through mathematical predictions"). Links are in the project readme.