Q.Find the equation for the ellipse that satisfies the given conditions: Vertices $(0, \pm 13)$, foci $(0, \pm 5)$.

Concept: Standard form of an ellipse with vertical major axis.

Since the vertices and foci lie on the $y$-axis, the major axis is vertical. The standard form is:

where $a > b > 0$.

Step 1: From vertices $(0, \pm 13)$, we have $a = 13$.

Step 2: From foci $(0, \pm 5)$, we have $c = 5$.

Step 3: Use the relationship $c^2 = a^2 - b^2$: …

Understanding the Standard Form

An ellipse is the set of all points whose distances from two fixed points (the foci) sum to a constant. When the ellipse is centered at the origin, its equation takes one of two standard forms depending on which axis is longer.

The key is to identify which axis contains the major diameter. The vertices are the endpoints of the major axis, and they're always farther from the center than any other point on the ellipse. The foci always lie on the major axis, between the center and the vertices.

Since both vertices $(0, \pm 13)$ and foci $(0, \pm 5)$ lie on the $y$-axis, the major axis is vertical. For a vertical major axis, the standard form is:

where $a > b > 0$, and the relationship $c^2 = a^2 - b^2$ connects the semi-major axis length $a$, semi-minor axis length $b$, and focal distance $c$.

Finding the Equation

1. Identify $a$ from the vertices

The vertices are at $(0, \pm 13)$, which means they're $13$ units from the center along the $y$-axis. This distance is the semi-major axis length:

2. Identify $c$ from the foci …