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

Moderate

A

$|\begin{array}{cc} bx + ay & cx + by \\ b' x + a' y & c' x + b' y \end{array}|$
B

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

$|\begin{array}{cc} bx + cy & ax + by \\ b' x + c' y & a' x + b' y \end{array}|$
D

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

Solution

Let $Δ = |\begin{array}{ccc} y^{2} & - xy & x^{2} \\ a & b & c \\ a' & b' & c' \end{array}|$
Then,
$\begin{array}{rcl} Δ & = & \frac{1}{xy} |\begin{array}{ccc} {xy}^{2} & - xy & x^{2} y \\ ax & b & cy \\ a' x & b' & c' y \end{array}| [Applying C_{1} \to {xC}_{1}, C_{3} \to {yC}_{3}] \\ = & \frac{1}{xy} |\begin{array}{ccc} 0 & - xy & 0 \\ ax + by & b & bx + cy \\ a' x + b' y & b' & b' x + c' y \end{array}| [Applying C_{1} \to C_{1} + {yC}_{2}, C_{3} \to C_{3} + {xC}_{2}] \\ = & \frac{1}{xy} xy |\begin{array}{cr} ax + by & bx + cy \\ a' x + b' y & b' x + c' y \end{array}| [Expanding along R_{1}] \\ = & |\begin{matrix} ax + by & bx + cy \\ a' x + b' y & b' x + c' y \end{matrix}| \end{array}$

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