Ellipse: (x/a)² + (y/b)² = 1

An ellipse centred at the origin with semi-major axis $a$ and semi-minor axis $b$ has equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Its eccentricity is $e = \sqrt{1 - b^2/a^2}$, with $0 \le e < 1$. When $a = b$, the ellipse becomes a circle.

Slide $a$ and $b$. The lab plots the ellipse and reports the foci at $(\pm c, 0)$ with $c = \sqrt{a^2 - b^2}$, plus eccentricity $e = c/a$. Set $a = 5$, $b = 3$ to see $c = 4$ and $e = 0.8$.

Planetary orbits are ellipses with the Sun at one focus (Kepler's first law). Whispering galleries — like the one in Bijapur's Gol Gumbaz — use elliptical domes so sound from one focus converges at the other. The lab's focus dots make this property visually obvious.

If the slider for $a$ falls below $b$, the y-axis becomes the major axis and the foci move to the y-axis: $c = \sqrt{b^2 - a^2}$. Students mechanically write $c = \sqrt{a^2 - b^2}$ and get a negative under the root. Always identify the larger denominator first.