Ans: A
Note that the $x$- and $y$-intercepts of the $12x-5y=60$ are $5$ and $-12$ respectively. Therefore, the coordinates of $A$ and $B$ are $(5,0)$ and $(0,-12)$ respectively. Let $(x,y)$ the coordinates of $P$. Hence, the equation of the locus of $P$ is
$\begin{array}{rcl}
AP & = & BP \\
\sqrt{(x-5)^2 +(y-0)^2} & = & \sqrt{(x-0)^2 +(y-(-12))^2} \\
x^2 -10x +25 +y^2 & = & x^2 +y^2 +24y +144 \\
10x +24y +119 & = & 0
\end{array}$
