Ellipse
Standard Equation and Elements
An ellipse is the set of points the sum of whose distances from two fixed points (the foci) is constant. A hyperbola is the set of points the difference of whose distances from two foci is constant. One word changes — sum versus difference — and a closed oval becomes a pair of opening branches.
Ellipse (foci on the $x$-axis, $a > b$):
Here $2a$ is the major axis and $2b$ the minor axis. The foci sit at $(\pm c, 0)$ where $c = \sqrt{a^2 - b^2}$, and the vertices (endpoints of the major axis) are $(\pm a, 0)$.
Hyperbola (foci on the $x$-axis):
The vertices are $(\pm a, 0)$, the foci are $(\pm c, 0)$ with $c = \sqrt{a^2 + b^2}$, and the two branches approach the slanted asymptotes $y = \pm \dfrac{b}{a}x$.
For both curves the shape is summarised by the eccentricity $e = \dfrac{c}{a}$, and both share the same latus-rectum formula:
| Parameter | Ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ | Hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ |
|---|---|---|
| Defining condition | sum of focal distances $= 2a$ | difference of focal distances $= 2a$ |
| Vertices | $(\pm a, 0)$ | $(\pm a, 0)$ |
| Foci | $(\pm c, 0)$ | $(\pm c, 0)$ |
| Relation | $c = \sqrt{a^2 - b^2}$ | $c = \sqrt{a^2 + b^2}$ |
| Eccentricity | $0 < e < 1$ | $e > 1$ |
| Latus rectum | $\dfrac{2b^2}{a}$ | $\dfrac{2b^2}{a}$ |
The lone difference in the $c$-relations — a minus for the ellipse, a plus for the hyperbola — is exactly what pushes the eccentricity below $1$ in one case and above $1$ in the other.
Deeper Insight — eccentricity is the dial that turns one conic into another: Every conic in this chapter is a slice of a double cone, and a single number, the eccentricity $e$, records how steeply the slicing plane is tilted. A circle is the perfectly level cut with $e = 0$; tilt a little and you get an ellipse with $0 < e < 1$; tilt until the plane runs parallel to the cone's side and the closed oval breaks open into a parabola with $e = 1$; tilt further still and you cut both nappes of the cone, producing a hyperbola with $e > 1$. This is why ellipses, parabolas and hyperbolas are not three unrelated curves but three settings of the same dial — and why $e$ appears in the focus-directrix description of all of them ($PF = e \cdot PM$). The sign flip in $c = \sqrt{a^2 \mp b^2}$ is the algebraic shadow of this geometry: for an ellipse the foci must sit inside, so $c < a$ and $e < 1$; for a hyperbola the foci lie beyond the vertices, so $c > a$ and $e > 1$. Hold on to the cone picture and the entire chapter reads as one story rather than a list of formulae.
- Here $a^2 = 25$, $b^2 = 9$, so $a = 5$, $b = 3$ (major axis along the $x$-axis).
- $c = \sqrt{a^2 - b^2} = \sqrt{25 - 9} = \sqrt{16} = 4$.
- Vertices: $(\pm 5, 0)$; foci: $(\pm 4, 0)$.
- Eccentricity $e = \dfrac{c}{a} = \dfrac{4}{5}$.
Answer: Vertices $(\pm 5, 0)$, foci $(\pm 4, 0)$, $e = \dfrac{4}{5}$.
- Here $a^2 = 16$, $b^2 = 9$, so $a = 4$, $b = 3$.
- For a hyperbola $c = \sqrt{a^2 + b^2} = \sqrt{16 + 9} = \sqrt{25} = 5$.
- Vertices: $(\pm 4, 0)$; foci: $(\pm 5, 0)$.
- Eccentricity $e = \dfrac{c}{a} = \dfrac{5}{4}$.
Answer: Vertices $(\pm 4, 0)$, foci $(\pm 5, 0)$, $e = \dfrac{5}{4}$.
- Here $a^2 = 36$, $b^2 = 20$, so $a = 6$.
- Length of latus rectum $= \dfrac{2b^2}{a} = \dfrac{2(20)}{6} = \dfrac{40}{6} = \dfrac{20}{3}$.
Answer: $\dfrac{20}{3}$.
- Vertices $(\pm a, 0)$ give $a = 6$, so $a^2 = 36$.
- Foci $(\pm c, 0)$ give $c = 4$, and $b^2 = a^2 - c^2 = 36 - 16 = 20$.
- The equation is $\dfrac{x^2}{36} + \dfrac{y^2}{20} = 1$.
Answer: $\dfrac{x^2}{36} + \dfrac{y^2}{20} = 1$.
- Here $a^2 = 9$, $b^2 = 4$, so $a = 3$, $b = 2$.
- The asymptotes of $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ are $y = \pm \dfrac{b}{a}x$.
- Substitute: $y = \pm \dfrac{2}{3}x$.
Answer: $y = \dfrac{2}{3}x$ and $y = -\dfrac{2}{3}x$.
- Foci $(\pm c, 0)$ give $c = 5$; eccentricity $e = \dfrac{c}{a} = \dfrac{5}{3}$ gives $a = \dfrac{c}{e} = \dfrac{5}{5/3} = 3$, so $a^2 = 9$.
- For a hyperbola $b^2 = c^2 - a^2 = 25 - 9 = 16$.
- The equation is $\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1$.
Answer: $\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1$.
- Ellipse: sum of focal distances is constant; $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ with $c=\sqrt{a^2-b^2}$ and $0
- Hyperbola: difference of focal distances is constant; $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ with $c=\sqrt{a^2+b^2}$ and $e>1$.
- Both have vertices $(\pm a,0)$, foci $(\pm c,0)$, eccentricity $e=\dfrac{c}{a}$ and latus rectum $\dfrac{2b^2}{a}$.
- Hyperbola asymptotes are $y=\pm\dfrac{b}{a}x$; the ellipse has none.
- All conics are cone sections; the eccentricity $e$ (0, <1, =1, >1) names circle, ellipse, parabola, hyperbola.
Eccentricity and Latus Rectum
b² = a²(1 − e²) gives the eccentricity e (0 < e < 1). The latus rectum length is 2b²/a.
- b² = a²(1 − e²), 0 < e < 1.
- Latus rectum = 2b²/a.
Foci and Properties
The foci are at (±ae, 0) and the sum of focal distances of any point equals 2a.
- Foci at (±ae, 0).
- Sum of focal distances = 2a.