giving a circle $c_1$ with radius $R_1$, Center$(X_1,Y_1)$ and a point $p$. (R1,X1,Y1 and p are known). I want to calculate $r_2$ and center point $x_2$,$y_2$ for circle $c_2$. $c_2$ covers the biggest possible area inside $c_1$ without containing point $p$ inside it.
As Oscar Lanzi writes in his comment, the circle that you’re looking for will be tangent to the outer circle at one end of a diameter $\overline{pq}$. A tangent line to a circle is perpendicular to the diameter at the point of tangency, so $(x_1,y_1)$ lies on $\overline{pq}$. The center of the inner circle $(x_2,y_2)$ is of course the midpoint of $\overline{pq}$, and $R_2$ is just half the length of $\overline{pq}$, i.e., $$R_2 = \frac12\left(\|(x_1,y_1)-p\|+R_1\right)$$ and $$(x_2,y_2) = p + R_2{(x_1,y_1)-p \over \|(x_1,y_1)-p\|} = {R_2 \over 2R_2-R_1}(x_1,y_1) + {R_2-R_1 \over 2R_2-R_1}p.$$