12.1 The Ellipse

An ellipse is a closed curve where the sum of the distances from any point on the curve to two fixed points (called foci) is constant. This definition drives everything about how ellipses are written, graphed, and applied. Mastering the standard form equations and key features here will set you up for the rest of analytic geometry.

Ellipse Equations and Graphing

Equations of ellipses in standard form

The standard form of an ellipse depends on two things: where the center is and which direction the longer axis runs. In every case, $a$ is the semi-major axis (the longer one) and $b$ is the semi-minor axis (the shorter one), so $a > b$ always holds.

Centered at the origin:

• Horizontal major axis: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ The larger denominator is under $x^2$, so the ellipse stretches farther left and right.
• Vertical major axis: $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ The larger denominator is under $y^2$, so the ellipse stretches farther up and down.

Centered at $(h, k)$:

• Horizontal major axis: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$
• Vertical major axis: $\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$

The quick rule: whichever denominator is bigger tells you the direction of the major axis. If the bigger number is under the $x$-term, the major axis is horizontal. If it's under the $y$-term, the major axis is vertical.

Parametric form (centered at the origin): $x = a \cos t$, $y = b \sin t$, where $0 \leq t < 2\pi$

Graphing ellipses with varied centers

1. Find the center $(h, k)$ from the equation and plot it. If there's no shift (no $h$ or $k$), the center is the origin.
2. Determine orientation. Compare the two denominators. The larger one corresponds to $a^2$, and the variable it sits under tells you horizontal vs. vertical.
3. Calculate $a$ and $b$ by taking the square root of each denominator. For example, if the equation is $\frac{(x-2)^2}{25} + \frac{(y+1)^2}{9} = 1$, then $a = 5$ and $b = 3$.
4. Plot the vertices along the major axis: $(h \pm a,\; k)$ for horizontal or $(h,\; k \pm a)$ for vertical.
5. Plot the co-vertices along the minor axis: $(h,\; k \pm b)$ for horizontal or $(h \pm b,\; k)$ for vertical.
6. Sketch the curve through all four points, keeping it smooth and symmetric about both axes through the center.

Key features of ellipses

Foci

The two foci sit along the major axis, each a distance $c$ from the center, where:

$c^2 = a^2 - b^2$

• Horizontal ellipse centered at $(h, k)$: foci at $(h \pm c,\; k)$
• Vertical ellipse centered at $(h, k)$: foci at $(h,\; k \pm c)$

The defining property of an ellipse is that for any point $P$ on the curve, the sum of distances to the two foci equals $2a$. This constant-sum property is what distinguishes an ellipse from other conic sections.

Vertices and co-vertices

• The two vertices are the endpoints of the major axis (the farthest points from the center).
• The two co-vertices are the endpoints of the minor axis (the closest points to the center along the perpendicular direction).

Eccentricity

Eccentricity measures how "stretched out" an ellipse is:

$e = \frac{c}{a}$

Since $c < a$ for any ellipse, eccentricity falls in the range $0 < e < 1$. An eccentricity close to 0 means the ellipse is nearly circular. An eccentricity close to 1 means it's very elongated. A perfect circle has $e = 0$ (because $c = 0$, meaning the two foci merge into one point at the center).

Additional Representations and Properties

• Polar form: An ellipse with one focus at the pole can be written as $r = \frac{a(1 - e^2)}{1 - e \cos \theta}$, where $e$ is the eccentricity. This form is especially useful in orbital mechanics.
• Auxiliary circles: The major auxiliary circle has radius $a$ and is centered at the ellipse's center (circumscribes the ellipse). The minor auxiliary circle has radius $b$ (inscribed within the ellipse). These are helpful for geometric constructions and deriving the parametric equations.
• Conjugate diameters: A pair of diameters where each is parallel to the tangent line at the endpoint of the other. In a circle, conjugate diameters are simply perpendicular diameters.

Real-world applications of ellipses

• Planetary orbits: Planets travel in elliptical paths with the Sun at one focus. Earth's orbit, for instance, has an eccentricity of about 0.017, making it nearly circular.
• Whispering galleries: Rooms with elliptical cross-sections (like the U.S. Capitol Building) reflect sound from one focus to the other, so a whisper at one focus can be heard clearly across the room.
• Architecture: Elliptical arches provide both structural stability and visual appeal. The dome of St. Peter's Basilica incorporates elliptical geometry.
• Engineering: Elliptical gears convert uniform rotation into variable-speed rotation, useful in certain machinery and mechanisms.