Multiple choice

The sum of the areas of two circles, which touch each other externally is $153$ $\pi$. If the sum of their radius is $15$, then the ratio of the larger to the smaller radius is

  1. $4:1$
  2. $2:1$
  3. $3:1$

  4. None of these
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A Correct answer
Explanation

Let radii be r1 and r2. We have r1 + r2 = 15 and pi(r1^2 + r2^2) = 153pi. Solving r1^2 + r2^2 = 153 and (r1 + r2)^2 = 225, we find 2*r1*r2 = 225 - 153 = 72, so r1*r2 = 36. The roots of x^2 - 15x + 36 = 0 are 12 and 3. The ratio is 12:3 = 4:1.

AI explanation

The area of a circle is pi times the square of its radius, so the sum of the areas of two circles is pi multiplied by the sum of their squared radii. Setting the sum of the squared radii to 153 gives the equations R squared plus r squared equals 153 and R plus r equals 15. Squaring the second equation yields R squared plus r squared plus 2 times R times r equals 225, and substituting the first equation gives 2 times R times r equals 72. Solving the system gives the larger radius as 12 and the smaller radius as 3, making their ratio 4 to 1.

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