If p≠a,q≠b,r≠cand the system of equations  px+ay+az=0,bx+qy+bz=0,cx+cy+rz=0 has a non-trivial solution, then the value of pp−a+qq−b+rr−c=_________

The condition for a homogeneous system to have a non-trivial solution is Δ=0⇒p   a   ab   q     bc    c     r=0   c3→c3−c1 &c2→c2−c1⇒p            a−p          a−pb            q−b               0c                 0                        r−c=0Expanding along C3we get(a-p)bq-bc0+(r-c)p-pb=0q-b=0(a-p)[-c(q-b)]+(r-c)[p(q-b)-b(a-p)]=0p−aq−bc+r−cq−bp+br−cp−a=0Dividing with p−aq−br−c we getcr−c+pp−a+bq−b=0⇒pp−a+qq−b+rr−c=2