If the foci of an ellipse are `(0,+-1)` and the minor axis is of unit length, then find the equation of the ellipse. The axes of ellipse are the coordinate axes.

Explanation

To find the equation of the ellipse given the foci and the length of the minor axis, we can follow these steps:

### Step 1: Identify the parameters of the ellipse
The foci of the ellipse are given as (0, ±1). This indicates that the foci are located on the y-axis, which means the major axis is vertical. The center of the ellipse is at the origin (0, 0).

### Step 2: Determine the values of 'b' and 'a'
Since the minor axis is of unit length, we know that the length of the minor axis (2a) is 1. Therefore, we can find 'a':
\[ 2a = 1 \implies a = \frac{1}{2} \]

The distance from the center to the foci (c) is given by the coordinates of the foci. Here, c = 1 (since the foci are at (0, ±1)).

### Step 3: Use the relationship between a, b, and c
For an ellipse, the relationship between a, b, and c is given by:
\[ c^2 = b^2 - a^2 \]
We know:
- \(c = 1\)
- \(a = \frac{1}{2}\)

Substituting these values into the equation:
\[ 1^2 = b^2 - \left(\frac{1}{2}\right)^2 \]
\[ 1 = b^2 - \frac{1}{4} \]
\[ b^2 = 1 + \frac{1}{4} = \frac{5}{4} \]

### Step 4: Write the equation of the ellipse
The standard form of the equation of an ellipse with a vertical major axis is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Substituting the values of \(a^2\) and \(b^2\):
\[ \frac{x^2}{\left(\frac{1}{2}\right)^2} + \frac{y^2}{\frac{5}{4}} = 1 \]
This simplifies to:
\[ \frac{x^2}{\frac{1}{4}} + \frac{y^2}{\frac{5}{4}} = 1 \]
Multiplying through by 4 to eliminate the denominators:
\[ 4 \cdot \frac{x^2}{\frac{1}{4}} + 4 \cdot \frac{y^2}{\frac{5}{4}} = 4 \]
This results in:
\[ 16x^2 + 4y^2 = 4 \]
Dividing the entire equation by 4 gives:
\[ 4x^2 + y^2 = 1 \]

### Final Equation
Thus, the equation of the ellipse is:
\[ 4x^2 + y^2 = 1 \]