Two circles touch each other externally. The sum of their areas is 58π cm² and the distance between their centres is 10 cm. Find the radii of the two circles.

Let r₁ and r₂ be the radii of the two circles.

Sum of the areas = 58π

⇒ A₁ + A₂ = 58π

⇒ πr₁² + πr₂² = 58π

⇒ π(r₁² + r₂²) = 58π

⇒ $\cancel{π}$ (r₁² + r₂²) = 58 $\cancel{π}$

⇒ r₁² + r₂² = 58 ……………(1)

Also, since the circles touch externally, the sum of their radii equals the distance between their centers.

r₁ + r₂ = 10

⇒ r₁ = 10 - r₂

Substituting the value of r₁ in equation (1),

⇒ (10 - r₂)² + r₂² = 58

⇒ 10² + r₂² - 2 x 10 x r₂ + r₂² = 58

⇒ 100 + 2r₂² - 20r₂ = 58

⇒ 100 + 2r₂² - 20r₂ - 58 = 0

⇒ 2r₂² - 20r₂ - 42 = 0

⇒ r₂² - 10r₂ - 21 = 0

⇒ r₂² - 3r₂ - 7r₂ - 21 = 0

⇒ r₂(r₂ - 3) - 7(r₂ - 3) = 0

⇒ (r₂ - 3)(r₂ - 7) = 0

⇒ r₂ = 3 cm or 7 cm

Using equation (1), we find r₁:

If r₂ = 3, then r₁ = 10 - 3 = 7 cm

If r₂ = 7, then r₁ = 10 - 7 = 3 cm

Hence, the radii of the two circles are 7 cm and 3 cm.