Find the eqation of the ellipse having foci (0,1),(0,-1) and minor axis of length 1.

Answer

To find the equation of the ellipse with foci at (0, 1) and (0, -1) and a minor axis of length 1, we can follow these steps:

### Step 1: Identify the center and orientation of the ellipse
The foci of the ellipse are given as (0, 1) and (0, -1). This indicates that the ellipse is centered at the origin (0, 0) and is oriented vertically along the y-axis.

**Hint:** The foci help determine the center and orientation of the ellipse.

### Step 2: Determine the values of \(c\) and \(b\)
The distance from the center to each focus is denoted as \(c\). Since the foci are at (0, 1) and (0, -1), we have:
\[ c = 1 \]

The minor axis length is given as 1, which means the semi-minor axis \(b\) is:
\[ 2b = 1 \implies b = \frac{1}{2} \]

**Hint:** The distance to the foci gives \(c\), and the length of the minor axis gives \(b\).

### Step 3: 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:
\[ 1^2 = a^2 - \left(\frac{1}{2}\right)^2 \]
\[ 1 = a^2 - \frac{1}{4} \]

**Hint:** This relationship is crucial for finding \(a\).

### Step 4: Solve for \(a^2\)
Rearranging the equation gives:
\[ a^2 = 1 + \frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4} \]

**Hint:** Make sure to combine fractions correctly.

### Step 5: Write the equation of the ellipse
The standard form of the equation of an ellipse centered at the origin with a vertical major axis is:
\[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \]

Substituting the values of \(a^2\) and \(b^2\):
\[ \frac{x^2}{\left(\frac{1}{2}\right)^2} + \frac{y^2}{\frac{5}{4}} = 1 \]
\[ \frac{x^2}{\frac{1}{4}} + \frac{y^2}{\frac{5}{4}} = 1 \]

Multiplying through by 4 to eliminate the denominators:
\[ 4 \cdot \frac{x^2}{\frac{1}{4}} + 4 \cdot \frac{y^2}{\frac{5}{4}} = 4 \]
\[ 16x^2 + 4y^2 = 4 \]

Dividing through by 4 gives:
\[ 4x^2 + y^2 = 1 \]

**Hint:** Always simplify the equation to its standard form.

### Final Answer
The equation of the ellipse is:
\[ 4x^2 + y^2 = 1 \]