Q.Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse $\frac{x^2}{4} + \frac{y^2}{25} = 1$.

Concept: Standard Form of an Ellipse

The given ellipse is $\frac{x^2}{4} + \frac{y^2}{25} = 1$, which has the form $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ with $a^2 = 25$ and $b^2 = 4$.

Since $a^2 > b^2$, the major axis lies along the $y$-axis. We have $a = 5$ and $b = 2$.

For this vertical ellipse, $c^2 = a^2 - b^2 = 25 - 4 = 21$, so $c = \sqrt{21}$.

Key elements:

• Vertices (endpoints of major axis): $(0, \pm a) = (0, \pm 5)$
• Foci: $(0, \pm c) = (0, \pm \sqrt{21})$
• Length of major axis: $2a = 10$
• Length of minor axis: $2b = 4$
• Eccentricity: $e = \frac{c}{a} = \frac{\sqrt{21}}{5}$
• Length of latus rectum: $\frac{2b^2}{a} = \frac{2 \cdot 4}{5} = \frac{8}{5}$

The standard form of an ellipse centered at the origin is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. The key insight is to identify which denominator is larger—that determines which axis is major. Here $25 > 4$, so the major axis lies along the $y$-axis. This means $b^2 = 25$ (giving $b = 5$) and $a^2 = 4$ (giving $a = 2$).

For an ellipse with major axis along the $y$-axis, the relationship between the semi-axes and the focal distance $c$ is:

This comes from the geometric definition: the sum of distances from any point on the ellipse to the two foci is constant and equals $2b$ (the length of the major axis).

Let me work through each element systematically.

1. Calculate the focal distance $c$

So $c = \sqrt{21}$.

2. Coordinates of the foci

Since the major axis is vertical, the foci lie on the $y$-axis at $(0, \pm c)$.

Foci: $(0, \sqrt{21})$ and $(0, -\sqrt{21})$

3. Coordinates of the vertices

The vertices are the endpoints of the major axis, located at $(0, \pm b)$.

Vertices: $(0, 5)$ and $(0, -5)$

4. Length of the major axis

The major axis spans from one vertex to the other, so its length is $2b$.

Length of major axis: $2 \times 5 = 10$

5. Length of the minor axis

The minor axis has length $2a$.

Length of minor axis: $2 \times 2 = 4$

6. Eccentricity $e$

Eccentricity measures how "stretched" the ellipse is, defined as:

where $b$ is the semi-major axis length.

7. Length of the latus rectum

The latus rectum is a chord through a focus, perpendicular to the major axis. Its length is given by: