Q.Find the equation for the ellipse that satisfies the given conditions: Length of major axis $26$, foci $(\pm 5, 0)$.

Concept: Locus of Point (Ellipse from foci and major axis length).

The foci are on the $x$-axis, so the ellipse is horizontal: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

Step 1: Length of major axis $= 2a = 26 \implies a = 13$.

Step 2: Foci at $(\pm c, 0) = (\pm 5, 0) \implies c = 5$. …

The key to any ellipse problem is understanding what the given numbers actually mean geometrically. The foci are at $(\pm 5, 0)$, which tells you two things immediately: the ellipse is centered at the origin (since the foci are symmetric about the origin), and the major axis is along the x-axis (because the foci lie on the x-axis). The length of the major axis is 26, so the semi-major axis $a$ is half of that: $a = 13$.

Now, for an ellipse centered at the origin with the major axis along the x-axis, the standard equation is:

where $a > b > 0$. The foci are at $(\pm c, 0)$, and the relationship between $a$, $b$, and $c$ is:

You already have $a = 13$ and $c = 5$. The only missing piece is $b$, and the formula above gives it directly.

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

   The length of the major axis is $2a = 26$, so $a = 13$.

2. Identify $c$ from the foci.

   The foci are $(\pm 5, 0)$, so $c = 5$.

3. Use the ellipse relationship to find $b^2$.

Hence $b = 12$ (since $b > 0$).

4. Write the equation. …