If $ax + by - 3 = 0$ is the equation of the shortest chord of the circle $\displaystyle \left ( x-3 \right )^{2}+\left ( y-4 \right )^{2}=4$ passing through the point $(2,3)$, then $|a + b|$ is

The shortest chord passing through (2,3) is the one perpendicular to the radius at (2,3). Center is (3,4). Radius vector from (3,4) to (2,3) has slope (3-4)/(2-3) = -1/-1 = 1. The chord must have slope -1. Equation: y - 3 = -1(x - 2) => y - 3 = -x + 2 => x + y - 5 = 0. Comparing to ax + by - 3 = 0, we need to scale to match constant term. Multiply by 3/5: (3/5)x + (3/5)y - 3 = 0. So a=3/5, b=3/5. |a+b| = 6/5.