Let \( y = x \) be the equation of a chord of the circle \( C_{1} \) (in the closed half-plane \( x \ge 0 \)) of diameter \( 10 \) passing through the origin. Let \( C_{2} \) be another circle described on the given chord as its diameter. If the equation of the chord of the circle \( C_{2} \), which passes through the point \( (2, 3) \) and is farthest from the center of \( C_{2} \), is \( x + a y + b = 0 \), then \( a - b \) is equal to:
