Two circles touch each other externally. The sum of their areas is 74π cm² and the distance between their centres is 12 cm. Find the diameters of the circle.

Given:

Sum of area = 740π cm²

The distance between their centres = 12 cm

Let $\mathrm{‘}{r}_1\mathrm{‘}$r_1 and $\mathrm{‘}{r}_2\mathrm{‘}$r_2 be the radius of 2 circles.

⇒ A₁ + A₂ = 74π

⇒ πr₁² + πr₂² = 74π

⇒ π(r₁² + r₂²) = 74π

⇒ $\cancel{π}$ (r₁² + r₂²) = 74 $\cancel{π}$

⇒ r₁² + r₂² = 74 ……………(1)

And, r₁ + r₂ = 12

⇒ r₁ = 12 - r₂

Putting the value of r₁ in equation (1),

⇒ (12 - r₂)² + r₂² = 74

⇒ 12² + r₂² - 2 x 12 x r₂ + r₂² = 74

⇒ 144 + 2r₂² - 24r₂ = 74

⇒ 144 + 2r₂² - 24r₂ - 74 = 0

⇒ 2r₂² - 24r₂ + 70 = 0

⇒ r₂² - 12r₂ + 35 = 0

⇒ r₂² - 7r₂ - 5r₂ + 35 = 0

⇒ r₂(r₂ - 7) - 5(r₂ - 7) = 0

⇒ (r₂ - 7)(r₂ - 5) = 0

⇒ r₂ = 7 cm or 5 cm

⇒ r₁ = 12 - r₂ = 12 - 7 or 12 - 5 = 5 cm or 7 cm

Hence, the radius of 2 circles = 7 cm and 5 cm.