Q.The equation of the ellipse having foci $(0, 1)$, $(0, -1)$ and minor axis of length 1 is ________.

The equation of an ellipse is determined by its center, the lengths of its semi-major and semi-minor axes, and its orientation.

1. The foci are given as $(0, 1)$ and $(0, -1)$. This means the center of the ellipse is the midpoint of the foci, which is $(0,0)$. The distance from the center to each focus is $c=1$. Since the foci lie on the y-axis, the major axis is along the y-axis.
2. The length of the minor axis is $2b=1$, so the semi-minor axis $b=1/2$.
3. For an ellipse, the relationship between the semi-major axis ($a$), semi-minor axis ($b$), and the distance from the center to a focus ($c$) is $a^2 = b^2 + c^2$. Substituting the values: $a^2 = (1/2)^2 + 1^2 = 1/4 + 1 = 5/4$. …

The equation of an ellipse is fundamentally derived from its definition as a locus of points. An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points (called foci) is constant. This constant sum is equal to the length of the major axis, denoted as $2a$.

The key to finding the equation of an ellipse is to identify its center, the lengths of its semi-major axis ($a$) and semi-minor axis ($b$), and its orientation (whether the major axis is horizontal or vertical). These parameters are interconnected by the relationship $b^2 = a^2 - c^2$, where $c$ is the distance from the center to each focus.

Let's break down the problem:

1. Identify the Foci and Center:

   The given foci are $F_1(0, 1)$ and $F_2(0, -1)$.

   The center of the ellipse is the midpoint of the segment connecting the foci.

   Center $C = \left(\frac{0+0}{2}, \frac{1+(-1)}{2}\right) = (0, 0)$.

   The distance between the foci is $2c$.

   $2c = \sqrt{(0-0)^2 + (1-(-1))^2} = \sqrt{0^2 + 2^2} = \sqrt{4} = 2$.

   Therefore, $c = 1$.

   Since the foci lie on the y-axis, the major axis of the ellipse is vertical. This means the standard form of the ellipse equation will be $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$.

2. Determine the Semi-minor Axis Length:

   The problem states that the minor axis has a length of 1.

   The length of the minor axis is $2b$.

   So, $2b = 1$, which implies $b = \frac{1}{2}$.

   Then, $b^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$.

3. Calculate the Semi-major Axis Length:

   We use the fundamental relationship between $a$, $b$, and $c$ for an ellipse:

   $b^2 = a^2 - c^2$

   Substitute the values we found for $b^2$ and $c$:

   $\frac{1}{4} = a^2 - 1^2$

   $\frac{1}{4} = a^2 - 1$

   $a^2 = 1 + \frac{1}{4}$

   $a^2 = \frac{5}{4}$. …