If `Delta_1=|(a,b,2c),(p,q,2r),(x,y,2z)| and Delta_2=|(r,2p,q),(2z,4x,2y),(c,2a,b)|` then `Delta_1//Delta_2` is equal to

Options

A. 1
B. 2
C. `-1`
D. `1//2`

Answer

To solve the problem, we need to find the ratio \( \frac{\Delta_1}{\Delta_2} \) where

\[ \Delta_1 = \begin{vmatrix} a & b & 2c \\ p & q & 2r \\ x & y & 2z \end{vmatrix} \]

and

\[ \Delta_2 = \begin{vmatrix} r & 2p & q \\ 2z & 4x & 2y \\ c & 2a & b \end{vmatrix} \]

### Step 1: Calculate \( \Delta_1 \)

Using the determinant formula for a 3x3 matrix:

\[ \Delta_1 = a \begin{vmatrix} q & 2r \\ y & 2z \end{vmatrix} - b \begin{vmatrix} p & 2r \\ x & 2z \end{vmatrix} + 2c \begin{vmatrix} p & q \\ x & y \end{vmatrix} \]

Calculating the 2x2 determinants:

1. \( \begin{vmatrix} q & 2r \\ y & 2z \end{vmatrix} = q \cdot 2z - 2r \cdot y = 2qz - 2ry \)

2. \( \begin{vmatrix} p & 2r \\ x & 2z \end{vmatrix} = p \cdot 2z - 2r \cdot x = 2pz - 2rx \)

3. \( \begin{vmatrix} p & q \\ x & y \end{vmatrix} = p \cdot y - q \cdot x = py - qx \)

Substituting these back into the equation for \( \Delta_1 \):

\[ \Delta_1 = a(2qz - 2ry) - b(2pz - 2rx) + 2c(py - qx) \]

Expanding this gives:

\[ \Delta_1 = 2aqz - 2ary - 2bpz + 2brx + 2c(py - qx) \]

### Step 2: Calculate \( \Delta_2 \)

Using the same determinant formula:

\[ \Delta_2 = r \begin{vmatrix} 4x & 2y \\ 2a & b \end{vmatrix} - 2p \begin{vmatrix} 2z & 2y \\ c & b \end{vmatrix} + q \begin{vmatrix} 2z & 4x \\ c & 2a \end{vmatrix} \]

Calculating the 2x2 determinants:

1. \( \begin{vmatrix} 4x & 2y \\ 2a & b \end{vmatrix} = 4x \cdot b - 2y \cdot 2a = 4xb - 4ay \)

2. \( \begin{vmatrix} 2z & 2y \\ c & b \end{vmatrix} = 2z \cdot b - 2y \cdot c = 2zb - 2yc \)

3. \( \begin{vmatrix} 2z & 4x \\ c & 2a \end{vmatrix} = 2z \cdot 2a - 4x \cdot c = 4az - 4xc \)

Substituting these back into the equation for \( \Delta_2 \):

\[ \Delta_2 = r(4xb - 4ay) - 2p(2zb - 2yc) + q(4az - 4xc) \]

Expanding this gives:

\[ \Delta_2 = 4rxb - 4ray - 4pbz + 4pyc + 4qaz - 4qxc \]

### Step 3: Find the ratio \( \frac{\Delta_1}{\Delta_2} \)

Now we have:

\[ \Delta_1 = 2aqz - 2ary - 2bpz + 2brx + 2c(py - qx) \]

\[ \Delta_2 = 4rxb - 4ray - 4pbz + 4pyc + 4qaz - 4qxc \]

Notice that both determinants can be factored.

Taking out common factors from \( \Delta_1 \) and \( \Delta_2 \):

\[ \Delta_1 = 2(A) \]

\[ \Delta_2 = 4(B) \]

Where \( A \) and \( B \) are the remaining expressions. Thus:

\[ \frac{\Delta_1}{\Delta_2} = \frac{2A}{4B} = \frac{1}{2} \cdot \frac{A}{B} \]

### Final Result

The final result simplifies to:

\[ \frac{\Delta_1}{\Delta_2} = \frac{1}{2} \]