In the michelson-morley experiment a null result or a net

Question

1. In the Michelson-Morley experiment, a null result, or a net fringe shift of ΔN = 0 was obtained when the Michelson interfero meter was rotated by 90°. This implies there is no path difference between the two arms of the interferometer: i.e., ΔL = 0. Consider the interfero meter when arm AC is oriented parallel to the direction of motion. Use equations (1.8) and (1.9) without applying the binomial approximation to show that the length of arm AC, L₂ , has contracted and is given by L₂ = L₁√1-β² , where β = v / c and v is the speed of the interferometer (and Earth) through the ether

2. Given the Lorentz transformation equations, x' = y(x - vt) and x = y(x' +yt), show that t' = y(t' - vx/c²), where y = 1/(1 v² /c²)^I/2