The length of the common chord of the circle $x^{2} + y^{2} + 4x + 6y + 4 = 0$ and $x^{2} + y^{2} + 6x 4y + 4 = 0$ is-

Subtracting the second circle equation from the first eliminates the x^2 and y^2 terms, producing the equation of the common chord as x - 5y = 0. The first circle x^2 + y^2 + 4x + 6y + 4 = 0 has its center at (-2, -3) and a radius squared of 9. Using the perpendicular distance formula from a point to a line, the distance from the center to the chord x - 5y = 0 is (-2 - (-15)) divided by the square root of 26, which is 13/sqrt(26). The length of the common chord is 2 times the square root of (9 minus 169/26), simplifying to 2 times the square root of (65/26), or sqrt(10).

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