Question

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

Answer 1

Let the radii of two circles be r₁ and r₂ respectively.
Sum of the areas of two circles = 130π sq. cm

⇒ πr₁² + πr₂² = 130π

⇒ r₁² + r₂² = 130 ….(i)

Also, distance between two radii = 14 cm

⇒ r₁ + r₂ = 14

⇒ r₁ = (14 - r₂)

Substituting the value of r₁ in (i), we get

(14 - r₂)² + r₂² = 130

⇒ 196 - 28r₂ + r₂² + r₂² = 130

⇒ 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 or r₂ = 3

⇒ r₁ = 14 - 11 = 3 or r₁ = 14 - 3 = 11

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