Find the equation of the ellipse whose centre is at (-2, 3) and whose semi-axes are 3 and 2, when the major axis is (i) parallel to the axes of x, (ii) parallel to the axis of y.

Answer

To find the equations of the ellipse whose center is at (-2, 3) and whose semi-axes are 3 and 2, we will consider two cases: (i) when the major axis is parallel to the x-axis, and (ii) when the major axis is parallel to the y-axis.

### Step-by-Step Solution:

**Case (i): Major Axis Parallel to the X-axis**

1. **Identify the center and semi-axes:**
- Center (h, k) = (-2, 3)
- Semi-major axis (a) = 3 (since it is greater than the semi-minor axis)
- Semi-minor axis (b) = 2

2. **Write the standard equation of the ellipse:**
The standard form of the equation of an ellipse centered at (h, k) with a horizontal major axis is:
\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]

3. **Substitute the values into the equation:**
\[ \frac{(x + 2)^2}{3^2} + \frac{(y - 3)^2}{2^2} = 1 \]
This simplifies to:
\[ \frac{(x + 2)^2}{9} + \frac{(y - 3)^2}{4} = 1 \]

4. **Multiply through by the least common multiple (LCM) to eliminate the denominators:**
Multiply by 36 (LCM of 9 and 4):
\[ 4(x + 2)^2 + 9(y - 3)^2 = 36 \]

5. **Expand the equation:**
\[ 4(x^2 + 4x + 4) + 9(y^2 - 6y + 9) = 36 \]
\[ 4x^2 + 16x + 16 + 9y^2 - 54y + 81 = 36 \]

6. **Combine like terms:**
\[ 4x^2 + 9y^2 + 16x - 54y + 61 = 0 \]

Thus, the equation of the ellipse when the major axis is parallel to the x-axis is:
\[ 4x^2 + 9y^2 + 16x - 54y + 61 = 0 \]

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**Case (ii): Major Axis Parallel to the Y-axis**

1. **Identify the center and semi-axes:**
- Center (h, k) = (-2, 3)
- Semi-major axis (b) = 3 (now it is greater than the semi-minor axis)
- Semi-minor axis (a) = 2

2. **Write the standard equation of the ellipse:**
The standard form of the equation of an ellipse centered at (h, k) with a vertical major axis is:
\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]

3. **Substitute the values into the equation:**
\[ \frac{(x + 2)^2}{2^2} + \frac{(y - 3)^2}{3^2} = 1 \]
This simplifies to:
\[ \frac{(x + 2)^2}{4} + \frac{(y - 3)^2}{9} = 1 \]

4. **Multiply through by the least common multiple (LCM) to eliminate the denominators:**
Multiply by 36 (LCM of 4 and 9):
\[ 9(x + 2)^2 + 4(y - 3)^2 = 36 \]

5. **Expand the equation:**
\[ 9(x^2 + 4x + 4) + 4(y^2 - 6y + 9) = 36 \]
\[ 9x^2 + 36x + 36 + 4y^2 - 24y + 36 = 36 \]

6. **Combine like terms:**
\[ 9x^2 + 4y^2 + 36x - 24y + 36 = 0 \]

Thus, the equation of the ellipse when the major axis is parallel to the y-axis is:
\[ 9x^2 + 4y^2 + 36x - 24y + 36 = 0 \]

### Summary of Results:
1. When the major axis is parallel to the x-axis:
\[ 4x^2 + 9y^2 + 16x - 54y + 61 = 0 \]
2. When the major axis is parallel to the y-axis:
\[ 9x^2 + 4y^2 + 36x - 24y + 36 = 0 \]