Two circles touch externally. The sum of their areas is 117π cm² and the distance between their centres is 15 cm. Find the radii of the two circles.

Let x and y be the radii of the of two circles with centers A and B respectively.

By formula,

Area of a circle = π (radius)²

Given,

Sum of areas = 117π cm²

πx² + πy² = 117π

x² + y² = 117 …….(1)

Given,

Distance between centers = 15 cm

x + y = 15

x = 15 - y …….(2)

Substituting value of x from equation (2) in (1), we get :

⇒ (15 - y)² + y² = 117

⇒ 225 - 30y + y² + y² = 117

⇒ 2y² - 30y + 225 - 117 = 0

⇒ 2y² - 30y + 108 = 0

⇒ 2(y² - 15y + 54) = 0

⇒ y² - 15y + 54 = 0

⇒ y² - 9y - 6y + 54 = 0

⇒ y(y - 9) - 6(y - 9) = 0

⇒ (y - 6)(y - 9) = 0

⇒ y - 6 = 0 or y - 9 = 0

⇒ y = 6 cm or y = 9 cm.

If y = 6 cm, then x = 15 - y = 9 cm.

If y = 9 cm, then x = 15 - y = 6 cm.

Since, x is the radius of circle with center A and from figure it is clear that A is the circle with greater radius thus, x = 9 cm and y = 6 cm.

Hence, the radii of two circles with centers A and B = 9 cm and 6 cm respectively.