Off the top of my head I'd guess to calculate tangent lines to the circle in the general form:

y = m*x + b(m)

where b(m) is your intercept as a function of m

I'm guessing there will be two general equations here, one for m>0 and one for m<0

then plug in your point to find m in each equation

watch out for m=0 and m=infinity though, as this method won't account for these cases

edit: This may or may not be correct, and may or may not be the easiest way