Find the transformed equations of the following when the origin is shifted to the point `(1,1)` by a translation of axes :
`(i) x^(2)+xy-3y^(2)-y+2=0`
`(ii) xy-y^(2)-x+y=0`
`(iii) xy-x-y+1=0`
`(iv) x^(2)-y^(2)-2x+2y=0`

Answer

To find the transformed equations when the origin is shifted to the point (1, 1) by a translation of axes, we will replace \(x\) with \(x' + 1\) and \(y\) with \(y' + 1\), where \(x'\) and \(y'\) are the new coordinates after the shift.

Let's solve each part step by step.

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

1. **Substitute \(x = x' + 1\) and \(y = y' + 1\)**:
\[ (x' + 1)^2 + (x' + 1)(y' + 1) - 3(y' + 1)^2 - (y' + 1) + 2 = 0 \]

2. **Expand the equation**:
\[ (x'^2 + 2x' + 1) + (x'y' + x' + y' + 1) - 3(y'^2 + 2y' + 1) - (y' + 1) + 2 = 0 \]
\[ x'^2 + 2x' + 1 + x'y' + x' + y' + 1 - 3y'^2 - 6y' - 3 - y' - 1 + 2 = 0 \]

3. **Combine like terms**:
\[ x'^2 + x'y' - 3y'^2 + (2x' + x' - 6y' - y') + (1 + 1 - 3 - 1 + 2) = 0 \]
\[ x'^2 + x'y' - 3y'^2 + 3x' - 6y' = 0 \]

**Transformed Equation**:
\[ x'^2 + x'y' - 3y'^2 + 3x' - 6y' = 0 \]

---

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

1. **Substitute \(x = x' + 1\) and \(y = y' + 1\)**:
\[ (x' + 1)(y' + 1) - (y' + 1)^2 - (x' + 1) + (y' + 1) = 0 \]

2. **Expand the equation**:
\[ (x'y' + x' + y' + 1) - (y'^2 + 2y' + 1) - (x' + 1) + (y' + 1) = 0 \]
\[ x'y' + x' + y' + 1 - y'^2 - 2y' - 1 - x' - 1 + y' + 1 = 0 \]

3. **Combine like terms**:
\[ x'y' - y'^2 + (x' - x' + y' - 2y' + y') + (1 - 1 - 1 + 1) = 0 \]
\[ x'y' - y'^2 = 0 \]

**Transformed Equation**:
\[ x'y' - y'^2 = 0 \]

---

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

1. **Substitute \(x = x' + 1\) and \(y = y' + 1\)**:
\[ (x' + 1)(y' + 1) - (x' + 1) - (y' + 1) + 1 = 0 \]

2. **Expand the equation**:
\[ x'y' + x' + y' + 1 - x' - 1 - y' - 1 + 1 = 0 \]
\[ x'y' + x' + y' - x' - y' = 0 \]

3. **Simplify**:
\[ x'y' = 0 \]

**Transformed Equation**:
\[ x'y' = 0 \]

---

### (iv) \(x^2 - y^2 - 2x + 2y = 0\)

1. **Substitute \(x = x' + 1\) and \(y = y' + 1\)**:
\[ (x' + 1)^2 - (y' + 1)^2 - 2(x' + 1) + 2(y' + 1) = 0 \]

2. **Expand the equation**:
\[ (x'^2 + 2x' + 1) - (y'^2 + 2y' + 1) - (2x' + 2) + (2y' + 2) = 0 \]
\[ x'^2 + 2x' + 1 - y'^2 - 2y' - 1 - 2x' - 2 + 2y' + 2 = 0 \]

3. **Combine like terms**:
\[ x'^2 - y'^2 + (2x' - 2x' + 2y' - 2y') + (1 - 1 - 2 + 2) = 0 \]
\[ x'^2 - y'^2 = 0 \]

**Transformed Equation**:
\[ x'^2 - y'^2 = 0 \]

---