Find the length of the chord intercepted by the circle ${\mathrm{x}}^2+{\mathrm{y}}^2-8\mathrm{x}-2\mathrm{y}-8=0$ on the line x + y + 1 = 0

Given circle is ${\mathrm{x}}^2+{\mathrm{y}}^2-8\mathrm{x}-2\mathrm{y}-8=0$
$\text{ Center }\mathrm{C}=(-\mathrm{g},-\mathrm{f})=(4,1)$
$\text{ Radius }\mathrm{r}=\sqrt{{\mathrm{g}}^2+{\mathrm{f}}^2-\mathrm{c}}=\sqrt{16+1+8}=5$

$\mathrm{d}={\perp }^{\text{er }}\text{ distance from centre }\mathrm{C}(4,1)\text{ to chord }$
x + y + 1 = 0
$=\frac{|4+1+1|}{\sqrt{1^2+1^2}}=\frac{6}{\sqrt{2}}=3\sqrt{2}$
$\mathrm{Length} \mathrm{of} \mathrm{the} \mathrm{chord} \mathrm{AB}=2\sqrt{{\mathrm{r}}^2-{\mathrm{d}}^2}$
$=2\sqrt{5^2-(3\sqrt{2}{)}^2}=2\sqrt{25-18}=2\sqrt{7}$