Multiple choice

If the circle $\mathrm{x}^{2}+\mathrm{y}^{2}+2\mathrm{g}\mathrm{x}+2\mathrm{f}\mathrm{y}+\mathrm{c}=0$ bisects the circumference of the circle $\mathrm{x}^{2}+\mathrm{y}^{2}+2\mathrm{g}{1} \mathrm{x}+2\mathrm{f}{1}\mathrm{y}+\mathrm{c}_{1}=0$, then the length of the common chord of the circles is

  1. $2\sqrt{g_{1}^{2}+f_{1}^{2}-c_{1}}$
  2. $\sqrt{g_{1}^{2}+f_{1}^{2}-c_{1}}$
  3. $\sqrt{g^{2}+f^{2}-c}$
  4. $2\sqrt{g^{2}+f^{2}-c}$
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A Correct answer
Explanation

If a circle bisects the circumference of another, the common chord is the diameter of the bisected circle. The equation of the common chord is found by subtracting the two circle equations: 2(g-g1)x + 2(f-f1)y + (c-c1) = 0. The length of the diameter of the second circle is 2 * sqrt(g1^2 + f1^2 - c1).

AI explanation

When one circle bisects the circumference of another, their common chord represents a diameter of the second circle. The length of a diameter of the circle x squared plus y squared plus 2g1 x plus 2f1 y plus c1 = 0 is twice its radius. Using the standard circle radius formula r = the square root of (g1 squared plus f1 squared minus c1), the common chord length is 2 times the square root of (g1 squared plus f1 squared minus c1).

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