Q.Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the latus rectum of the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$.

The given ellipse is $\frac{x^2}{25} + \frac{y^2}{9} = 1$. Since $25 > 9$, the major axis is along the $x$-axis.

Here $a^2 = 25 \Rightarrow a = 5$, and $b^2 = 9 \Rightarrow b = 3$.

For an ellipse, $c^2 = a^2 - b^2 = 25 - 9 = 16$, so $c = 4$.

• Vertices: $(\pm a, 0) = (\pm 5, 0)$
• Foci: $(\pm c, 0) = (\pm 4, 0)$
• Length of major axis: $2a = 10$
• Length of minor axis: $2b = 6$
• Eccentricity: $e = \frac{c}{a} = \frac{4}{5}$
• Latus rectum: $\frac{2b^2}{a} = \frac{2 \cdot 9}{5} = \frac{18}{5}$

The equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$ is already in the standard form of an ellipse centred at the origin. The standard form is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ when the major axis is horizontal, and $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ when it is vertical — the larger denominator always belongs to $a^2$, the semi-major axis squared.

Here $25 > 9$, so $a^2 = 25$ and $b^2 = 9$. That means $a = 5$ and $b = 3$. Since the larger number is under $x^2$, the major axis lies along the x-axis. This immediately tells us the vertices are on the x-axis and the foci are also on the x-axis.

The relationship that ties everything together for an ellipse is $c^2 = a^2 - b^2$, where $c$ is the distance from the centre to each focus. Let’s compute it:

Now we have all the numbers we need. Let’s list each required quantity step by step.

1. Foci: For a horizontal major axis, the foci are at $(\pm c, 0)$. So the foci are $(-4, 0)$ and $(4, 0)$.

2. Vertices: The vertices are the endpoints of the major axis, at $(\pm a, 0)$. So the vertices are $(-5, 0)$ and $(5, 0)$.

3. Length of major axis: This is simply $2a = 2 \times 5 = 10$.

4. Length of minor axis: This is $2b = 2 \times 3 = 6$.

5. Eccentricity: $e = \frac{c}{a} = \frac{4}{5}$. Eccentricity tells us how “stretched” the ellipse is — closer to 0 means more circular, closer to 1 means more elongated. Here $0.8$ is fairly elongated.

6. Latus rectum: The latus rectum of an ellipse is a chord through a focus perpendicular to the major axis. Its length is given by $\frac{2b^2}{a}$. So: