[UPSC CSE 1990] Special Theory of Relativity (Q418, 20M)

Write down Lorentz transformation relations and prove that x^2 + y^2 + z^2 - c^2 t^2 is invariant under this transformation. An event occurs at x^1 = 60\text{ m} at t^1 = 8 \times 10^{-8}\text{ s} in a reference frame S' which is moving along the common x or x' axis with a speed 3c/5 with reference to a stationary from S. The origins of the two frames coincide at t=0, t'=0. Deduce the space time coordinates of the event in the frame S.