If |a−b p−q x−y ; b−c q−r y−z ; c−a r−p z−x| = k|a b c ; p q r ; x y z|, then what is the value of k?

Add all three rows of the LHS determinant: (a−b)+(b−c)+(c−a) = 0, similarly for other columns. So the sum of rows is zero, meaning the rows are linearly dependent. Hence the determinant equals 0, giving k = 0.