The length of the common chord of the two circles $ x^2+y^2-4y=0$ and $x^2+y^2-8x-4y+11=0$ is

Subtracting the first circle's equation from the second gives the equation of the common chord: 8x + 11 = 0, so x = -11/8. Substituting x = -11/8 into the first circle, x^2 + y^2 - 4y = 0, yields y^2 - 4y + 121/64 = 0. The length of the chord cut by this line on the first circle, which has center (0, 2) and radius 2, is given by the formula 2 root(r^2 - d^2), where d is the perpendicular distance from the center to the chord x = -11/8. Calculating the distance gives d = 11/8, and using the formula results in 2 root(4 - 121/64) = 2 root(135/64) = root 135 / 4.

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