Multiple choice

The length of the chord of circle $x^{2}+y^{2}-6x-16=0$ which is at a distance of $3cm$ from center

  1. $8$
  2. $5\sqrt{3}$
  3. $5\sqrt{3}/2$
  4. $none\ of\ these$
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A Correct answer
Explanation

Equation: (x-3)^2 + y^2 = 16 + 9 = 25. Center is (3,0), radius is 5. Distance from center to chord is 3. Half-length of chord = sqrt(5^2 - 3^2) = sqrt(16) = 4. Total length = 2 * 4 = 8.

AI explanation

The center of the circle is found by completing the square, giving (-3, 0) and a radius squared of 9 + 16 = 25, so the radius is 5. The relationship between the chord length L, the radius r, and the perpendicular distance d from the center to the chord is given by the formula L = 2 sqrt(r^2 - d^2). Substituting the known values gives L = 2 sqrt(5^2 - 3^2) = 2 sqrt(25 - 9) = 2 sqrt(16) = 8. The result is 8.

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