Multiple choice

Two circles touch each other externally. The sum of their area is $58$ $\pi\, cm^{2}$ and the distance between their center is $10\ cm$. Find the radii of the two circles.

  1. $7\ cm $and $3\ cm$
  2. $5\ cm $and $7\ cm$
  3. $3\ cm $and $2\ cm$
  4. $None\ of\ these$
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A Correct answer
Explanation

Let radii be r1 and r2. r1 + r2 = 10 (distance between centers). Area sum: pi*r1^2 + pi*r2^2 = 58*pi => r1^2 + r2^2 = 58. (r1+r2)^2 - 2*r1*r2 = 58 => 100 - 2*r1*r2 = 58 => 2*r1*r2 = 42 => r1*r2 = 21. Roots of t^2 - 10t + 21 = 0 are 7 and 3.

AI explanation

Let the radii of the two circles be r1 and r2. Since the circles touch externally, the distance between their centers gives the equation r1 + r2 = 10. The sum of their areas gives the equation pi(r1^2 + r2^2) = 58pi. Squaring the first equation gives (r1 + r2)^2 = 100, which expands to r1^2 + r2^2 + 2(r1)(r2) = 100. Substituting the sum of the squares gives 58 + 2(r1)(r2) = 100, so the product of the radii is 21. The radii are therefore the roots of x^2 - 10x + 21 = 0, which yields values of 7 cm and 3 cm.

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