Let a circle with center O And radius R.
let
another circle inside the first circle  with center o’ and radius r .
A/Q,
Area of 1st circle + area of 2nd circle = 116π cm²
⇒ πR² + πr² = 116π
⇒ π(R² + r²)  = 116π
⇒ R² + r² =116 ——————–(i)
Now,
Distance between the centers of circles = 6 cm
i.e, R – r = 6
⇒  R = r + 6  ——————-(ii)
From Eqn (i) & (ii),
(r + 6)² + r² = 116
⇒ r² + 12r +36 + r² =116
⇒ 2r² +12r +36 -116 = 0
⇒ 2r² +12r – 80 = 0
⇒ r² +6r – 40 = 0
⇒ r² +10r – 4r – 40 = 0
⇒ r(r + 10) – 4(r + 10) = 0
⇒ (r + 10)(r – 4) = 0
hence  r = 4 cm
r ≠ -10 cm        {∵ length can’t be -ve}
Therefore radii of the circles are
r = 4 cm ,
R = 4 + 6 = 10 cm.