If a hyperbola passing through the origin has $3x-4y-1=0$ and $4x-3y-6=0$ as its asymptotes, then the equations of its transverse and conjugate axes, are

The transverse axis is the bisector of the angle between asymptotes containing the origin and the conjugate axis is the other bisector. So, their equations are given by

$\frac{-3x+4y+1}{\sqrt{9+16}}=\frac{-4x+3y+6}{\sqrt{16+9}}$

and,

$\frac{-3x+4y+1}{\sqrt{9+16}}=-\frac{-4x+3y+6}{\sqrt{16+9}}$

$\Rightarrow$ $x+y-5=0$ and $x-y-1=0\text{. }$