The determinant $\begin{vmatrix}y^2 & -xy & x^2\\a & b & c\\a' & b' & c'\end{vmatrix} $ is equal to

$\begin{vmatrix}ax+by & bx+cy\\a'x+b'y & b'x+c'y\end{vmatrix} $

The correct answer is option (2) : $\begin{vmatrix}ax+by & bx+cy\\a'x+b'y & b'x+c'y\end{vmatrix} $

Let $Δ=\begin{vmatrix}y^2 & -xy & x^2\\a & b & c\\a' & b' & c'\end{vmatrix} $. Then,

$Δ=\frac{1}{xy}\begin{vmatrix}xy^2 & -xy & x^2y\\ax & b & cy\\a'x & b' & c'y'\end{vmatrix} $      $\begin{bmatrix} Applying\\C_1→C_1(x), C_3→C_3(y)\end{bmatrix} $

$=\frac{1}{xy}\begin{vmatrix}0 & -xy & 0\\ax+by & b & bx+cy\\a'x+b'y & b' & b'x+c'y'\end{vmatrix}$ $\begin{bmatrix} Applying \,\, C_1→C_1+yC_2,\\C_3→C_3+xC_2\end{bmatrix}$

$=\frac{1}{xy}.xy \begin{vmatrix}ax+by & bx+cy\\a'x+b'y & b'x+c'y\end{vmatrix} $  [Expanding alone $R_1$]

$=\begin{vmatrix}ax+by & bx+cy\\a'x+b'y & b'x+c'y\end{vmatrix} $