The length of the common chord of the circles $x^2 + y^2 = 12$ and $x^2 + y^2 - 4x + 3y - 2 = 0$, is
- $4\sqrt{2}$
- $5 \sqrt{2}$
- $2 \sqrt{2}$
- $6 \sqrt{2}$
The common chord equation is S1 - S2 = 0. (x^2 + y^2 - 12) - (x^2 + y^2 - 4x + 3y - 2) = 0, which is 4x - 3y - 10 = 0. Distance from center (0,0) to chord is |4(0) - 3(0) - 10| / sqrt(4^2 + 3^2) = 10/5 = 2. Chord length = 2 * sqrt(r^2 - d^2) = 2 * sqrt(12 - 4) = 2 * sqrt(8) = 4 * sqrt(2).
Subtracting the second circle's equation from the first gives the common chord equation 4x - 3y - 10 = 0. The center of the first circle x^2 + y^2 = 12 is (0,0) and its radius is sqrt(12). Using the perpendicular distance from a point to a line formula, the distance from the center to the chord is |4(0) - 3(0) - 10| / sqrt(4^2 + (-3)^2) = 10/5 = 2. Applying the geometric property that half a chord equals sqrt(radius^2 - distance^2), we get half-length as sqrt(12 - 4) = sqrt(8). The full length of the common chord is 2 * sqrt(8) = 4 * sqrt(2).