Two circles touch externally. The sum of their areas is 130π sq. cm and the distance between their centres is 14 cm. Find the radii of the circles.

Given:

Sum of areas of two circles = 130π cm²

The distance between their centres = 14 cm

Let $r$1 and $r2$ be the radii of 2 circles.

Area of the first circle + Area of the second circle = 130π

⇒ A₁ + A₂ = 130π

⇒ πr₁² + πr₂² = 130π

⇒ π(r₁² + r₂²) = 130π

⇒ $\cancel{π}$ (r₁² + r₂²) = 130 $\cancel{π}$

⇒ r₁² + r₂² = 130 ……………(1)

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

r₁ + r₂ = 14

⇒ r₁ = 14 - r₂

Putting the value of r₁ in equation (1),

⇒ (14 - r₂)² + r₂² = 130

⇒ 14² + r₂² - 2 x 14 x r₂ + r₂² = 130

⇒ 196 + 2r₂² - 28r₂ = 130

⇒ 196 + 2r₂² - 28r₂ - 130 = 0

⇒ 2r₂² - 28r₂ + 66 = 0

⇒ r₂² - 14r₂ + 33 = 0

⇒ r₂² - 11r₂ - 3r₂ + 33 = 0

⇒ r₂(r₂ - 11) - 3(r₂ - 11) = 0

⇒ (r₂ - 11)(r₂ - 3) = 0

⇒ r₂ = 11 cm or 3 cm

Using equation (1), we find r₁:

If r₂ = 11, then r₁ = 14 - 11 = 3cm

If r₂ = 3, then r₁ = 14 - 3 = 11cm

Hence, the radii of the two circles are 11 cm and 3 cm.