The determinant ` |{:(y^(2),,-xy,,x^(2)),(a,,b ,,c),(a',,b',,c'):}|` is equal to

Options

A. `|(bx + ay,cx +by),(b'x + a'y,c' x + b' y)|`
B. `|(ax + by,bx + cy),(a'x + b'y,b' x + c' y)|`
C. `|(bx + cy,ax + by),(b' x + c'y,d' x + b' y)|`
D. none of these

Answer

B

Explanation

Let `Delta = |(y^(2),-xy,x^(2)),(a,b,c),(a',b',c')|`. Then,
`Delta = (1)/(xy) |(xy^(2),-xy,x^(2)y),(ax,b,cy),(a'x,b',c'y)| " "[("Applying"),(C_(1) rarr (x)"," C_(3) rarr C_(3) (y))]`
`= (1)/(xy) |(0,-xy,0),(ax + by,b,bx + cy),(a' x + b' y,b',b' x + c' y)| [("Applying" C_(1) rarr C_(1) + y C_(2)),(" " C_(3) rarr C_(3) + x C_(2))]`
`= (1)/(xy).xy |(ax + by,bx + cy),(a'x + b'y,b' x + c' y)|` [Expanding along `R_(1)`]
`= |(ax + by,bx + cy),(a' x + b' y,b' x + c' y)|`