Multiple choice

There are two orthogonal circles with radii $5$ and $12$ units, then the length of their common chord is?

  1. $\dfrac{60}{13}$
  2. $\dfrac{120}{13}$
  3. $\dfrac{30}{13}$
  4. $\dfrac{240}{13}$
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B Correct answer
Explanation

For orthogonal circles with radii r1=5 and r2=12, the distance between centers d = sqrt(r1^2 + r2^2) = sqrt(25 + 144) = 13. The length of the common chord is 2 * (r1 * r2) / d = 2 * (5 * 12) / 13 = 120/13.

AI explanation

For orthogonal circles, the square of the distance between their centers equals the sum of the squares of their radii, so d^2 = 5^2 + 12^2 = 169, giving d = 13. Using the formula for the length of a common chord, L = 2*r1*r2/d. Substituting the values gives L = 2*5*12/13 = 120/13. The length of the common chord is 120/13.

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