Conic Sections: Step-by-Step Worked Examples (4)

Question

Find the equation of the ellipse whose foci are at $(\pm 4, 0)$ and the length of the minor axis is $6$. Also find the eccentricity and the directrices.

Solution — Step by Step

Why This Works

The relation $c^2 = a^2 - b^2$ comes directly from the geometric definition of an ellipse — the sum of distances from any point to the two foci equals $2a$. Combined with the rule that minor axis = $2b$ and foci sit on the major axis at distance $c$, every ellipse is fully determined by any two of $\{a, b, c\}$.

Eccentricity measures how far the ellipse is from being circular. $e = 0$ is a circle; $e \to 1$ is a flattened ellipse approaching a parabola.

Alternative Method

Use the focal-radius definition. The sum of distances from any point $(x, y)$ on the ellipse to the foci $(\pm 4, 0)$ equals $2a$. From the minor-axis endpoint $(0, 3)$:

$\sqrt{16 + 9} + \sqrt{16 + 9} = 2 \times 5 = 10 \implies 2a = 10$

So $a = 5$ directly.

Common Mistake