Let y=x be the equation of a chord of the circle C_1 (in the closed half-plane x ≥ 0 ) of diameter 10 passing through the origin. Let C_2 be another…

Let $\displaystyle y=x$ be the equation of a chord of the circle $\displaystyle \mathrm{C}_1$ (in the closed half-plane $\displaystyle x \geq 0$ ) of diameter $\displaystyle 10$ passing through the origin. Let $\displaystyle \mathrm{C}_2$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle $\displaystyle \mathrm{C}_2$, which passes through the point $\displaystyle (2,3)$ and is farthest from the center of $\displaystyle \mathrm{C}_2$, is $\displaystyle x+a y+b=0$, then $\displaystyle a-b$ is equal to