If p a, q b, r c and the system of equations aligned & p x+a y+a z=0 & b x+q y+b z=0 & c x+c y+r z=0 aligned has a non-trivial solution, then the value of p p-a + q q-b + r r-c is

Step-by-step solution

As the given system of equations has a non-trivial solution. = | array lll p & a & a b & q & b c & c & r array |=0 Applying C₂ C₂-C₁ and C₃ C₃-C₁ = | array ccc p & a-p & a-p b & q-b & 0 c & 0 & r-c array |=0 Expanding along C₃ , we get (a-p) | array cc b & q-b c & 0 array |+(r-c) | array cc p & a-p b & q-b array |=0 aligned & (a-p)(-c)(q-b)+(r-c) & p(q-b)-b(a-p) =0 & & (p-a)(q-b) c+p(r-c)(q-b)+b(r-c)(p-a)=0 aligned Dividing by (p-a)(q-b)(r-c) , we get c r-c + p p-a + b q-b =0 p p-a + q q-b + r r-c = q-b q-b + r-c r