Which of the following is an equation of the ellipse centered at `(-2, 3)` with a minor axis of 4 parallel to the to the x - axis and a major axis of 6 parallel to the y - axis ?

Options

A. `((x-2)^(2))/(4)+((y-3)^(2))/(9)=1`
B. `((x+2)^(2))/(4)+((y-3)^(2))/(9)=1`
C. `((x-2)^(2))/(4)+((y+3)^(2))/(9)=1`
D. `((x+2)^(2))/(4)+((y+3)^(2))/(9)=1`

Explanation

To find the equation of the ellipse centered at \((-2, 3)\) with a minor axis of 4 (parallel to the x-axis) and a major axis of 6 (parallel to the y-axis), we can follow these steps:

### Step 1: Identify the center and axes lengths
The center of the ellipse is given as \((p, q) = (-2, 3)\). The lengths of the axes are given as:
- Minor axis (parallel to x-axis) = 4, which means \(2a = 4\) (where \(a\) is half the length of the minor axis).
- Major axis (parallel to y-axis) = 6, which means \(2b = 6\) (where \(b\) is half the length of the major axis).

### Step 2: Calculate values of \(a\) and \(b\)
From the lengths of the axes:
- \(2a = 4 \Rightarrow a = \frac{4}{2} = 2\)
- \(2b = 6 \Rightarrow b = \frac{6}{2} = 3\)

### Step 3: Write the standard form of the ellipse equation
The standard form of the equation of an ellipse centered at \((p, q)\) is given by:
\[ \frac{(x - p)^2}{a^2} + \frac{(y - q)^2}{b^2} = 1 \]
Substituting the values of \(p\), \(q\), \(a\), and \(b\):
- \(p = -2\)
- \(q = 3\)
- \(a = 2\) so \(a^2 = 4\)
- \(b = 3\) so \(b^2 = 9\)

### Step 4: Substitute the values into the equation
Substituting these values into the standard form:
\[ \frac{(x - (-2))^2}{2^2} + \frac{(y - 3)^2}{3^2} = 1 \]
This simplifies to:
\[ \frac{(x + 2)^2}{4} + \frac{(y - 3)^2}{9} = 1 \]

### Step 5: Final equation of the ellipse
Thus, the equation of the ellipse is:
\[ \frac{(x + 2)^2}{4} + \frac{(y - 3)^2}{9} = 1 \]

### Conclusion
The correct option that represents this equation is option 2.

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