Two circles touch internally. The sum of their areas is 170π cm² and the distance between their centres is 4 cm. Find the radii of the circles.

Let R be the radius of bigger circle and r be the radius of smaller circle.

Given,

Distance between centres (R - r) = 4 cm

Sum of areas = 170π cm²

If two circles touch internally then,

Distance between centres = R - r

∴ R - r = 4…..(1)

Given,

Sum of their areas = 170π

⇒ πR² + πr² = 170π

⇒ π(R² + r²) = 170π

⇒ R² + r² = 170

Substituting the value R = r + 4

⇒ (r + 4)² + r² = 170

⇒ r² + 16 + 2 × r × 4 + r² = 170

⇒ 2r² + 8r + 16 - 170 = 0

⇒ 2r² + 8r - 154 = 0

⇒ r² + 4r - 77 = 0

⇒ r² + 11r - 7r - 77 = 0

⇒ r(r + 11) - 7(r + 11) = 0

⇒ (r - 7)(r + 11) = 0

⇒ r = 7 cm.

Substituting r = 7 in equation (1), we get :

R - 7 = 4

R = 4 + 7 = 11 cm.

Hence, radii of the two circles = 7 cm and 11 cm.