The equation of hyperbola whose asymptotes are  $3x-4y+7=0$  and  $4x+3y+1=0$  which passes through the point (0,1) is $12x^2-7xy-12y^2+31x+17y+A=0$  and the equation of conjugate hyperbola is  $12x^2-7xy-12y^2+ax+by+c=0$, then the sum of the digits of  $a+b+c$  is ____

The equation of asymptotes are $3x-4y+7=0$  and  $4x+3y+1=0$.
The combined equation of asymptotes is  $(3x-4y+7)(4x+3y+1)=0$
or  $12x^2-7xy-12y^2+31x+17y+7=\mathrm{0......}\left(i\right)$
Let the equation of hyperbola be
 $12x^2-7xy-12y^2+31x+17y+A=\mathrm{0......}\left(ii\right)$
Since,(i) passes through (0,1)
$\therefore A=-5$ 
$\therefore$ Equation of hyperbola is  
$\therefore$ Equation of conjugate hyperbola=2(combined equation of asymptotes)-(equation of hyperbola)
$\Rightarrow$  Equation of conjugate hyperbola
$=2(12x^2-7xy-12y^2+31x+17y+7)-(12x^2-7xy-12y^2+31x+17y-5)=0$ 
 $\Rightarrow 12x^2-7xy-12y^2+31x+17y+19=\mathrm{0......}\left(iii\right)$
We have  $a=31,b=17,c=19$
 $\Rightarrow a+b+c=31+17+19=67$
$\therefore$  Required sum=6+7=13