If ` a = x + y , b = x - y , c = x + 2y ` then ` a^(2) + b^(2) + c^(2) - ab - bc - ca` is

Options

A. `4y^(2)`
B. `5y^(2)`
C. `6y^(2)`
D. `7y^(2)`

Answer

To solve the expression \( a^2 + b^2 + c^2 - ab - bc - ca \) given \( a = x + y \), \( b = x - y \), and \( c = x + 2y \), we will follow these steps:

### Step 1: Calculate \( a^2 \), \( b^2 \), and \( c^2 \)

1. **Calculate \( a^2 \)**:
\[ a^2 = (x + y)^2 = x^2 + 2xy + y^2 \]

2. **Calculate \( b^2 \)**:
\[ b^2 = (x - y)^2 = x^2 - 2xy + y^2 \]

3. **Calculate \( c^2 \)**:
\[ c^2 = (x + 2y)^2 = x^2 + 4xy + 4y^2 \]

### Step 2: Calculate \( ab \), \( bc \), and \( ca \)

1. **Calculate \( ab \)**:
\[ ab = (x + y)(x - y) = x^2 - y^2 \]

2. **Calculate \( bc \)**:
\[ bc = (x - y)(x + 2y) = x^2 + 2xy - y^2 - 2y^2 = x^2 + 2xy - 3y^2 \]

3. **Calculate \( ca \)**:
\[ ca = (x + 2y)(x + y) = x^2 + xy + 2xy + 2y^2 = x^2 + 3xy + 2y^2 \]

### Step 3: Substitute into the expression

Now substitute \( a^2 \), \( b^2 \), \( c^2 \), \( ab \), \( bc \), and \( ca \) into the expression \( a^2 + b^2 + c^2 - ab - bc - ca \):

\[ a^2 + b^2 + c^2 = (x^2 + 2xy + y^2) + (x^2 - 2xy + y^2) + (x^2 + 4xy + 4y^2) \]

Combine the terms:

\[ = 3x^2 + (2xy - 2xy + 4xy) + (y^2 + y^2 + 4y^2) = 3x^2 + 4xy + 6y^2 \]

Now substitute \( ab \), \( bc \), and \( ca \):

\[ -ab - bc - ca = -(x^2 - y^2) - (x^2 + 2xy - 3y^2) - (x^2 + 3xy + 2y^2) \]

Combine the terms:

\[ = -x^2 + y^2 - x^2 - 2xy + 3y^2 - x^2 - 3xy - 2y^2 \]
\[ = -3x^2 - 5xy + 2y^2 \]

### Step 4: Combine everything

Now, combine the results from Step 3:

\[ 3x^2 + 4xy + 6y^2 - (3x^2 + 5xy - 2y^2) \]

This simplifies to:

\[ 3x^2 + 4xy + 6y^2 - 3x^2 - 5xy + 2y^2 = (4xy - 5xy) + (6y^2 + 2y^2) = -xy + 8y^2 \]

### Final Result

Thus, the expression \( a^2 + b^2 + c^2 - ab - bc - ca \) simplifies to:

\[ \boxed{2y^2 - xy} \]