Q.Find the equation of the ellipse, whose length of the major axis is $20$ and foci are $(0, \pm 5)$.

The key idea is that the foci lie on the $y$-axis, so the major axis is vertical. The standard form of such an ellipse is

where $a > b > 0$, the distance from the centre to each focus is $c$, and $c^2 = a^2 - b^2$.

Step 1: The length of the major axis is $2a = 20$, so $a = 10$.

Step 2: The foci are at $(0, \pm 5)$, so $c = 5$. …

The standard form of an ellipse is your starting point. But which orientation? The foci are at $(0, \pm 5)$ — that tells you two things at once: the centre is at the origin, and the major axis is vertical (along the y-axis). When the major axis is vertical, the ellipse equation is:

where $a > b > 0$, $2a$ is the length of the major axis, and $c$ is the distance from the centre to each focus, related by $c^2 = a^2 - b^2$.

Now let’s work through the numbers.

1. Find $a$ from the major axis length.

   The problem says the length of the major axis is $20$. That means $2a = 20$, so $a = 10$.

   Therefore $a^2 = 100$.

2. Identify $c$ from the foci.

   Foci are at $(0, \pm 5)$, so the distance from the centre to each focus is $c = 5$.

   Hence $c^2 = 25$. …