Find what the following equations become when the origin is shifted to the point (1, 1)(i) `x^2+xy-3y^2-y+2=0`(ii) `xy-y^2-x+y=0`(iii) `xy-x-y+1=0`

Explanation

To find what the given equations become when the origin is shifted to the point (1, 1), we will replace the coordinates \( x \) and \( y \) with \( x + 1 \) and \( y + 1 \) respectively. This means that the new coordinates will be \( X = x - 1 \) and \( Y = y - 1 \).

Let's solve each equation step by step.

### Part (i): \( x^2 + xy - 3y^2 - y + 2 = 0 \)

1. **Substitute the new coordinates**:
\[ x = X + 1, \quad y = Y + 1 \]
Substitute these into the equation:
\[ (X + 1)^2 + (X + 1)(Y + 1) - 3(Y + 1)^2 - (Y + 1) + 2 = 0 \]

2. **Expand the equation**:
\[ (X^2 + 2X + 1) + (XY + X + Y + 1) - 3(Y^2 + 2Y + 1) - (Y + 1) + 2 = 0 \]
Simplifying gives:
\[ X^2 + 2X + 1 + XY + X + Y + 1 - 3Y^2 - 6Y - 3 - Y - 1 + 2 = 0 \]

3. **Combine like terms**:
\[ X^2 - 3Y^2 + XY + (2X + X) + (Y - 6Y) + (1 + 1 - 3 - 1 + 2) = 0 \]
This simplifies to:
\[ X^2 - 3Y^2 + XY + 3X - 6Y = 0 \]

### Part (ii): \( xy - y^2 - x + y = 0 \)

1. **Substitute the new coordinates**:
\[ (X + 1)(Y + 1) - (Y + 1)^2 - (X + 1) + (Y + 1) = 0 \]

2. **Expand the equation**:
\[ XY + X + Y + 1 - (Y^2 + 2Y + 1) - X - 1 + Y + 1 = 0 \]
Simplifying gives:
\[ XY + X + Y + 1 - Y^2 - 2Y - 1 - X + Y + 1 = 0 \]

3. **Combine like terms**:
\[ XY - Y^2 + (X - X) + (Y - 2Y + Y) + 1 - 1 + 1 = 0 \]
This simplifies to:
\[ XY - Y^2 = 0 \]

### Part (iii): \( xy - x - y + 1 = 0 \)

1. **Substitute the new coordinates**:
\[ (X + 1)(Y + 1) - (X + 1) - (Y + 1) + 1 = 0 \]

2. **Expand the equation**:
\[ XY + X + Y + 1 - X - 1 - Y - 1 + 1 = 0 \]
Simplifying gives:
\[ XY + X - X + Y - Y + 1 - 1 - 1 + 1 = 0 \]

3. **Combine like terms**:
\[ XY = 0 \]

### Final Results:
1. For \( x^2 + xy - 3y^2 - y + 2 = 0 \), the transformed equation is:
\[ X^2 - 3Y^2 + XY + 3X - 6Y = 0 \]

2. For \( xy - y^2 - x + y = 0 \), the transformed equation is:
\[ XY - Y^2 = 0 \]

3. For \( xy - x - y + 1 = 0 \), the transformed equation is:
\[ XY = 0 \]