The equation of the ellipse , with axes parallel to the coordinates axes , whose eccentricity is `(1)/(3)` and foci at (2,-2) and (2,4) is

Options

A. `((x-1)^(2))/(8)+((Y-2)^(2))/(9)=9`
B. `((x-2)^(2))/(8)+((Y-1)^(2))/(9)=9`
C. `((x-1)^(2))/(9)+((Y-2)^(2))/(8)=9`
D. `((x-2)^(2))/(9)+((Y-2)^(2))/(8)=9`

Explanation

To find the equation of the ellipse with the given properties, we will follow these steps:

### Step 1: Identify the foci and center of the ellipse
The foci of the ellipse are given as (2, -2) and (2, 4). The center of the ellipse can be found by calculating the midpoint of the line segment joining the two foci.

**Midpoint formula**:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Using the foci (2, -2) and (2, 4):
\[ \text{Center} = \left( \frac{2 + 2}{2}, \frac{-2 + 4}{2} \right) = \left( 2, 1 \right) \]

### Step 2: Determine the distance between the foci
The distance between the foci is given by the formula \(2c\), where \(c\) is the distance from the center to each focus. The distance between the foci can be calculated as:
\[ \text{Distance} = 4 - (-2) = 6 \]
Thus, \(2c = 6\) implies \(c = 3\).

### Step 3: Use the eccentricity to find \(b\)
The eccentricity \(e\) is given as \(\frac{1}{3}\). The relationship between \(c\), \(a\), and \(b\) in an ellipse is given by:
\[ e = \frac{c}{a} \]
Thus, we can express \(c\) in terms of \(a\):
\[ \frac{1}{3} = \frac{3}{a} \implies a = 9 \]

### Step 4: Relate \(a\), \(b\), and \(c\)
We know that:
\[ c^2 = a^2 - b^2 \]
Substituting the values we have:
\[ 3^2 = 9^2 - b^2 \implies 9 = 81 - b^2 \implies b^2 = 81 - 9 = 72 \]

### Step 5: Write the equation of the ellipse
Since the foci are vertically aligned, the standard form of the ellipse is:
\[ \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 \]
Where \((h, k)\) is the center of the ellipse. Substituting \(h = 2\), \(k = 1\), \(a^2 = 81\), and \(b^2 = 72\):
\[ \frac{(x - 2)^2}{72} + \frac{(y - 1)^2}{81} = 1 \]

### Final Equation
Thus, the equation of the ellipse is:
\[ \frac{(x - 2)^2}{72} + \frac{(y - 1)^2}{81} = 1 \]

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