Multiple choice

Length of the common chord of the circles $\displaystyle \left ( x-1 \right )^{2}+\left ( y+1 \right )^{2}=c^{2}: : and: : \left ( x+1 \right )^{2}+\left ( y-1 \right )^{2}=c^{2}$ is

  1. $\displaystyle \frac{1}{2}\sqrt{c^{2}-2}$
  2. $\displaystyle \sqrt{c^{2}-2}$
  3. $\displaystyle 2\sqrt{c^{2}-2}$
  4. $(c + 2)$
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C Correct answer
Explanation

The centers of the circles are (1, -1) and (-1, 1). The distance between centers is sqrt(2^2 + (-2)^2) = sqrt(8) = 2*sqrt(2). The common chord length formula for two circles of radius c is 2*sqrt(c^2 - d^2/4), where d is the distance between centers. Substituting d = 2*sqrt(2), we get 2*sqrt(c^2 - 8/4) = 2*sqrt(c^2 - 2).

AI explanation

Subtracting the second circle equation from the first gives 4x - 4y = 0, meaning the common chord lies on the line y = x. Substituting y = x into the first circle yields (x - 1)^2 + (x + 1)^2 = c^2, which simplifies to 2x^2 + 2 = c^2. Solving for x gives x = sqrt((c^2 - 2)/2), and since y = x, the distance between the two intersection points is 2 * sqrt(2) * sqrt((c^2 - 2)/2). This simplifies to 2 sqrt(c^2 - 2).

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