The problem of survival of a Brownian particle diffusing on a disk with a reflective boundary that has two absorbing arcs is treated analytically. The framework of boundary homogenization is applied to calculate the effective trapping rate of the disk boundary, and this enables estimation of the mean first passage time. The method of conformal mapping is applied to transform the original system to a simpler geometrical configuration (a flat reflective boundary with a periodic configuration of identical absorbing strips) for which the analytical solution is known. The expression for the mean first passage time is simplified for some limiting cases (small arc or small gap). The derived analytical expressions compare favorably with the results of Brownian particle simulations and other analytical results from the literature.