Q.Find the equation for the ellipse that satisfies the given conditions: Foci $(\pm 3, 0)$, $a = 4$.

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

The foci lie on the $x$-axis at $(\pm 3, 0)$, so the major axis is horizontal and the ellipse is centered at the origin. For such an ellipse, the standard equation is

where $a$ is the semi-major axis and $b$ is the semi-minor axis.

Step 1: We are given $a = 4$, so $a^2 = 16$.

Step 2: The foci are at $(\pm c, 0)$ where $c = 3$. The fundamental relationship for an ellipse is $c^2 = a^2 - b^2$. …

Understanding the Ellipse from Its Foci

An ellipse is the locus of all points for which the sum of distances to two fixed points (the foci) is constant. That constant sum equals $2a$, where $a$ is the semi-major axis. The foci lie along the major axis, and their distance from the center determines how "stretched" the ellipse is.

Since the foci are at $(\pm 3, 0)$, they lie on the $x$-axis, which tells us the major axis is horizontal. The center is at the origin $(0, 0)$, midway between the foci.

Finding the Equation Step by Step

1. Identify the given information

We have:

• Foci at $F_1 = (-3, 0)$ and $F_2 = (3, 0)$, so the focal distance from center is $c = 3$
• Semi-major axis length $a = 4$

2. Determine the orientation

Because the foci are on the $x$-axis, the major axis is horizontal. The standard form for such an ellipse centered at the origin is:

where $a > b$ (since the major axis is along $x$).

3. Use the fundamental relationship

For any ellipse, the parameters $a$, $b$, and $c$ satisfy:

This comes from the geometry: the semi-minor axis $b$ is shorter than $a$ by exactly the amount needed to place the foci at distance $c$ from the center.

4. Calculate $b^2$

Substituting our values: …