The T=S1 method gives the equation of a chord with a given midpoint. But to find the locus of the midpoint when the chord satisfies a condition (passes through a fixed point, subtends a given angle, etc.):

1. Let the midpoint be (h,k).
2. Write the chord equation using T=S1: for circle x^{2+y}^2=$r^2$, the chord with midpoint (h,k) is hx+ky=h^{2+k}^2.
3. Apply the additional condition to this chord.
4. Obtain a relation in h,k.
5. Replace h->x, k->y.

Example: Chord of x^{2+y}^2=25 subtending 90 degrees at center. The chord's distance from center = r*cos(45) = 5/sqrt(2). So h^{2+k}^2 = 25/2. Locus: x^{2+y}^2=25/2.