Question

Two circles intersect. The sum of their areas is 130 π sq. cm and the distance between their centers is 14 cm. Find the radius of the circles.​
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Answer 1

Answer:

The radii of two circles are 11 cm and 3 cm.

Step-by-step explanation:

Given :

Two circles intersect. The sum of their areas is 130 π sq. cm and the distance between their centers is 14 cm.

To find :

the radii of the circles

Solution :

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

• Area of the circle = πr²

Sum of the areas = 130π cm²

πr₁² + πr₂² = 130π

π(r₁² + r₂²) = 130π

r₁² + r₂² = 130 ➟ [1]

The distance between the centers of the two circles will be equal to the sum of their radii.

r₁ + r₂ = 14 cm ➟ [2]

Squaring equation [2],

(r₁ + r₂)² = 14²

r₁² + r₂² + 2r₁r₂ = 196

130 + 2r₁r₂ = 196  [ ∵ r₁² + r₂² = 130 ]

 2r₁r₂ = 196 - 130

 2r₁r₂ = 66

  r₁r₂ = 66/2

  r₁r₂ = 33

Now,

(r₁ - r₂)² = r₁² + r₂² - 2r₁r₂

(r₁ - r₂)² = 130 - 2(33)

(r₁ - r₂)² = 130 - 66

(r₁ - r₂)² = 64

(r₁ - r₂) = √64

(r₁ - r₂) = 8 ➟ [3]

Adding both equations [2] and [3],

r₁ + r₂ + r₁ - r₂ = 14 + 8

 2r₁ = 22

 r₁ = 22/2

 r₁ = 11 cm

Substitute in equation [3],

11 - r₂ = 8

r₂ = 11 - 8

r₂ = 3 cm

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