Multiple choice

Two circles touch each other externally. The sum of their areas is $130 \pi \, \ \text{cm}^2$ and the distance between their centres is $14$ cm. Find the diameters of the circle.

  1. $22, 8$
  2. $22, 6$
  3. $26, 8$
  4. $22, 4$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let radii be r1, r2. r1+r2 = 14 (distance between centers). Area sum: pi*r1^2 + pi*r2^2 = 130*pi. r1^2 + r2^2 = 130. (r1+r2)^2 - 2*r1*r2 = 130. 196 - 2*r1*r2 = 130. 2*r1*r2 = 66. r1*r2 = 33. Solving r^2 - 14r + 33 = 0 gives r=11 and r=3. Diameters are 22 and 6.

AI explanation

Let the radii of the two circles be r and R. The distance between their centers gives the equation r + R = 14. The sum of their areas gives the equation pi(r^2 + R^2) = 130 pi, which simplifies to r^2 + R^2 = 130. Using the identity (r + R)^2 = r^2 + R^2 + 2rR, we get 14^2 = 130 + 2rR, so 2rR = 66. The radii are the roots of x^2 - 14x + 33 = 0, which are 11 cm and 3 cm. The diameters are therefore twice these values, yielding 22 cm and 6 cm.

Keep practicing — related questions