Multiple choice

A circle of radius $3$ is placed at the centre of a circle of radius $r > 3$ such that the length of a chord of the larger circle tangent to the smaller circle is $8$. The probability that point randomly selected from inside of the larger circle lies inside the annular region of the two circles is

  1. $\dfrac{3}{4}$
  2. $\dfrac{9}{25}$
  3. $\dfrac{16}{25}$
  4. $\dfrac{7}{3}$
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C Correct answer
Explanation

Radius of small circle = 3. Chord length = 8. Distance from center to chord = sqrt(r^2 - 4^2). This distance is the radius of the small circle, so 3 = sqrt(r^2 - 16). 9 = r^2 - 16, so r^2 = 25, r=5. Area of large circle = 25pi. Area of small circle = 9pi. Annular region = 25pi - 9pi = 16pi. Probability = 16pi / 25pi = 16/25.

AI explanation

The length of the chord of the larger circle tangent to the smaller circle is 8, so half of the chord is 4. Using the property that the radius of the smaller circle is 3, we find the radius R of the larger circle using the Pythagorean theorem: R squared equals 3 squared plus 4 squared, which gives R squared equals 25. The area of the larger circle is 25 times pi, and the area of the smaller circle is 9 times pi. The probability that a randomly selected point lies in the annular region is the difference in areas divided by the larger area, which is (25 minus 9) divided by 25, resulting in 16 divided by 25.

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