Q.Find the length of the chord intercepted by the circle $x^2+y^2-x+3y-22=0$ on the line $y=x-3$.

Find the circle's centre/radius, the perpendicular distance from the centre to the line, then use $\text{chord length}=2\sqrt{r^2-d^2}$.

Circle: $x^2+y^2-x+3y-22=0$. Here $2g=-1\Rightarrow g=-\tfrac12$, $2f=3\Rightarrow f=\tfrac32$, $c=-22$.

Centre $=(-g,-f)=\left(\tfrac12,-\tfrac32\right)$

Radius $r=\sqrt{g^2+f^2-c}=\sqrt{\tfrac14+\tfrac94+22}=\sqrt{\tfrac{49}{2}}=\dfrac{7}{\sqrt2}$

Line: $y=x-3$, i.e. $x-y-3=0$.

Perpendicular distance from centre to the line: …