An ellipse, with foci at (0, 2) and (0, –2) and minor axis of length 4, passes through which of the following points?

Options

A. `(1,2sqrt(2))`
B. `(2,sqrt(2))`
C. `(2,2sqrt(2))`
D. `(sqrt(2),2)`

Explanation

To solve the problem of determining which point an ellipse with foci at (0, 2) and (0, -2) and a minor axis of length 4 passes through, we can follow these steps:

### Step 1: Identify the parameters of the ellipse
The foci of the ellipse are located at (0, 2) and (0, -2), indicating that the major axis is vertical (along the y-axis). The minor axis is horizontal.

### Step 2: Determine the lengths of the axes
The length of the minor axis is given as 4. This means:
\[ 2b = 4 \implies b = 2 \]
where \( b \) is the semi-minor axis length.

### Step 3: Find the distance of the foci from the center
The distance from the center of the ellipse to each focus (c) is given by the coordinates of the foci:
\[ c = 2 \]

### Step 4: Use the relationship between a, b, and c
For an ellipse, the relationship between the semi-major axis \( a \), semi-minor axis \( b \), and the distance to the foci \( c \) is given by:
\[ c^2 = a^2 - b^2 \]
Substituting the known values:
\[ 2^2 = a^2 - 2^2 \implies 4 = a^2 - 4 \implies a^2 = 8 \implies a = \sqrt{8} = 2\sqrt{2} \]

### Step 5: Write the equation of the ellipse
The standard form of the equation of an ellipse with a vertical major axis is:
\[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \]
Substituting the values of \( a^2 \) and \( b^2 \):
\[ \frac{x^2}{4} + \frac{y^2}{8} = 1 \]

### Step 6: Check which points satisfy the equation
Now we need to check which of the given points satisfy the equation of the ellipse. We will substitute each point into the equation.

1. **Point (1, 2√2)**:
\[ \frac{1^2}{4} + \frac{(2\sqrt{2})^2}{8} = \frac{1}{4} + \frac{8}{8} = \frac{1}{4} + 1 = \frac{5}{4} \quad (\text{not equal to } 1) \]

2. **Point (2, √2)**:
\[ \frac{2^2}{4} + \frac{(\sqrt{2})^2}{8} = \frac{4}{4} + \frac{2}{8} = 1 + \frac{1}{4} = \frac{5}{4} \quad (\text{not equal to } 1) \]

3. **Point (2, 2√2)**:
\[ \frac{2^2}{4} + \frac{(2\sqrt{2})^2}{8} = \frac{4}{4} + \frac{8}{8} = 1 + 1 = 2 \quad (\text{not equal to } 1) \]

4. **Point (√2, 2)**:
\[ \frac{(\sqrt{2})^2}{4} + \frac{2^2}{8} = \frac{2}{4} + \frac{4}{8} = \frac{1}{2} + \frac{1}{2} = 1 \quad (\text{equal to } 1) \]

### Conclusion
The ellipse passes through the point \( (\sqrt{2}, 2) \).