If $B (1, 3)$ is equidistant from $A (6, -1)$ and $C (x, 8)$, then $x =$

$\textbf{Step 1: Apply distance formula between the points.}$

$\text{Given that, B(1,3) is equidistant from the points }\mathrm{A(6,-1)\text{ and }C(x,8).}$

$\text{We know, distance between }\mathrm{(x_1,y_1)\ and\ (x_2,y_2)=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}}$

$\text{Distance between point B and point A,}$

$\Rightarrow\mathrm{BA=}\sqrt{(6-1)^2+(-1-3)^2}$

$=\sqrt{5^2+(-4)^2}$

$=\sqrt{25+16}$

$=\sqrt{41}.......(1)$

$\text{Distance between point B and point C,}$

$\Rightarrow\mathrm{BC=}\sqrt{(x-1)^2+(8-3)^2}$

$=\sqrt{x^2+1-2x+25}$

$=\sqrt{x^2-2x+26}......(2)$

$\textbf{Step 2: Apply suitable condition.}$

$\text{Since, B(1,3) is equidistant from the points }\mathrm{A(6,-1)\text{ and }C(x,8).}$

$\mathrm{\therefore BC=BA}$

$\mathrm{\Rightarrow BC^2=BA^2}$

$\Rightarrow x^2-2x+26=41$ $\textbf{[From (1) and (2)]}$

$\Rightarrow x^2-2x-15=0$

$\Rightarrow x^2-5x+3x-15=0$

$\Rightarrow (x-5)(x+3)=0$

$\Rightarrow x=5\text{ or }x=-3$

$\textbf{Hence, the correct option is B.}$