An application of elliptic integrals (Q1206863)

Suppose that \(X\) is a random variable having an arcsine distribution on the line segment between \((-1,0)\) and \((+1,0)\) of the Euclidean plane, and that \(D\) is the distance between \((X,0)\) and a fixed point \((p,q)\) on this plane. Then, the value of \(D\) and \(1/D\) can be expressed by three complete elliptic integrals in the Legendre forms.