Multiple choice

Two circles of radii $3, 4$ intersect orthogonally. Then the length of the common chord is

  1. $\dfrac {12}5$
  2. $\dfrac {24}{25}$
  3. $\dfrac {24}5$
  4. $\dfrac{25}{24}$
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C Correct answer
Explanation

For two circles of radii r1 and r2 intersecting orthogonally, the distance between centers d = sqrt(r1^2 + r2^2) = sqrt(3^2 + 4^2) = 5. The length of the common chord is (2 * r1 * r2) / d = (2 * 3 * 4) / 5 = 24/5.

AI explanation

Let the circles intersect at point A. The radii to A form a right angle, so triangle O1AO2 is a right triangle with hypotenuse 5. The altitude to this hypotenuse is the distance from A to the line of centers, calculated as 3 times 4 divided by 5, giving 12/5. The total common chord length is twice this altitude, resulting in 24/5.

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