The longest tune-out wavelength for potassium atoms, λ zero = 768.9701(4) nm, was measured using an atom interferometer with a large irradiance gradient supported in a multipass optical cavity. Systematic errors in λ zero measurements that arise from laser light, Doppler shifts, and the Earth's rotation are described. The ratio of oscillator strengths for the potassium D2 and D1 lines inferred from this λ zero measurement is ρ = f D2 /f D1 = 2.0066(11), and the ratio of line strengths is R = S D2 /S D1 = 1.9977(11).