Multiple choice

If the lengths of the chords intercepted by the circle ${x}^{2}+{y}^{2}+2gx+2fy=0$ from the coordinate axes are $10$ and $24$ units, respectively, then the radius of the circle is

  1. $17$
  2. $9$
  3. $14$
  4. $13$
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D Correct answer
Explanation

The length of the intercept on the x-axis is 2 * sqrt(g^2 - c) = 10, so g^2 - c = 25. The length of the intercept on the y-axis is 2 * sqrt(f^2 - c) = 24, so f^2 - c = 144. For a circle passing through the origin (as given by the equation), c=0. Thus g^2 = 25 and f^2 = 144. Radius = sqrt(g^2 + f^2 - c) = sqrt(25 + 144) = sqrt(169) = 13.

AI explanation

The circle intercepts lengths of 2g and 2f on the coordinate axes. Setting these equal to 10 and 24 gives g = 5 and f = 12. Using the radius formula r = sqrt(g^2 + f^2), we have r = sqrt(5^2 + 12^2) = sqrt(169) = 13. The result is 13.

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