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

Options

A. (a) `|{:(bx+ay,,cx+by),(b'x+a'y,,c'x+b'y):}|`
B. (b)`|{:(ax+by,,bx+cy),(a'x+b'y,,b'x+c'y):}|`
C. (c) `|{:(bx+cy,,ax+by),(b'x+c'y,,a'x+b'y):}|`
D. (d)`|{:(ax+by,,bc+cy),(a'x+b'y,,b'x+c'y):}|`

Explanation

To solve the determinant \( D = \begin{vmatrix} y^2 & -xy & x^2 \\ a & b & c \\ a' & b' & c' \end{vmatrix} \), we will follow the steps outlined below:

### Step 1: Define the Determinant
Let \( D = \begin{vmatrix} y^2 & -xy & x^2 \\ a & b & c \\ a' & b' & c' \end{vmatrix} \).

### Step 2: Apply Column Operations
We can simplify the determinant by performing column operations. Specifically, we will multiply the first column by \( x \) and the third column by \( y \). This gives us:

\[ D = \frac{1}{xy} \begin{vmatrix} xy^2 & -xy & yx^2 \\ a & b & c \\ a' & b' & c' \end{vmatrix} \]

### Step 3: Rewrite the Determinant
Now, we can rewrite the determinant as:

\[ D = \frac{1}{xy} \begin{vmatrix} xy^2 & -xy & yx^2 \\ a & b & c \\ a' & b' & c' \end{vmatrix} \]

### Step 4: Perform Row Operations
Next, we will perform row operations to simplify the determinant further. We can add \( \frac{y}{x} \) times the second column to the first column:

\[ D = \frac{1}{xy} \begin{vmatrix} xy^2 + yb & -xy & yx^2 \\ a + \frac{y}{x}b & b & c \\ a' + \frac{y}{x}b' & b' & c' \end{vmatrix} \]

### Step 5: Simplify the First Row
We can simplify the first row further. The first entry becomes \( xy^2 + yb = y(xy + b) \):

\[ D = \frac{1}{xy} \begin{vmatrix} y(xy + b) & -xy & yx^2 \\ a + \frac{y}{x}b & b & c \\ a' + \frac{y}{x}b' & b' & c' \end{vmatrix} \]

### Step 6: Factor Out \( y \)
Factoring out \( y \) from the first column gives:

\[ D = \frac{y}{xy} \begin{vmatrix} xy + b & -x & x^2 \\ a + \frac{y}{x}b & b & c \\ a' + \frac{y}{x}b' & b' & c' \end{vmatrix} \]

### Step 7: Final Determinant
Now we can express the determinant as:

\[ D = \frac{1}{x} \begin{vmatrix} xy + b & -x & x^2 \\ a + \frac{y}{x}b & b & c \\ a' + \frac{y}{x}b' & b' & c' \end{vmatrix} \]

### Step 8: Evaluate the Determinant
Now we can evaluate the determinant using the properties of determinants. The final result will yield:

\[ D = \begin{vmatrix} ax + by & -x & x^2 \\ a & b & c \\ a' & b' & c' \end{vmatrix} \]

### Conclusion
Thus, the value of the determinant is:

\[ D = \begin{vmatrix} ax + by & -x & x^2 \\ a & b & c \\ a' & b' & c' \end{vmatrix} \]