10.1 The Ellipse

Ellipse Equation and Graphing

An ellipse is a set of all points where the sum of the distances to two fixed points (called foci) is constant. Understanding the equation of an ellipse lets you identify its shape, orientation, size, and position on a coordinate plane.

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Standard Form of Ellipse Equations

For an ellipse centered at the origin, the standard form is:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

• $a$ is the semi-axis length in the horizontal direction (distance from center to the ellipse along the x-axis)
• $b$ is the semi-axis length in the vertical direction (distance from center to the ellipse along the y-axis)

Whichever denominator is larger tells you the direction of the major axis (the longer one). If $a > b$, the ellipse is wider than it is tall (horizontal major axis). If $b > a$, it's taller than it is wide (vertical major axis). When $a = b$, you just have a circle.

For an ellipse centered at $(h, k)$, the equation shifts accordingly:

$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

The $(h, k)$ values translate the entire ellipse so its center sits at that point instead of the origin. Notice that the signs inside the parentheses are subtracting $h$ and $k$, so if you see $\frac{(x-3)^2}{25} + \frac{(y+2)^2}{9} = 1$, the center is at $(3, -2)$, not $(3, 2)$.

Graphing Ellipses with Various Centers

Ellipse centered at the origin $(0, 0)$:

1. Identify $a^2$ and $b^2$ from the denominators, then take their square roots to get $a$ and $b$.
2. Plot the center at $(0, 0)$.
3. From the center, move $a$ units left and right to plot $(a, 0)$ and $(-a, 0)$.
4. From the center, move $b$ units up and down to plot $(0, b)$ and $(0, -b)$.
5. Sketch a smooth oval through all four points.

Ellipse centered at $(h, k)$:

1. Identify $a$, $b$, $h$, and $k$ from the equation.
2. Plot the center at $(h, k)$.
3. From the center, move $a$ units left and right to plot $(h+a, k)$ and $(h-a, k)$.
4. From the center, move $b$ units up and down to plot $(h, k+b)$ and $(h, k-b)$.
5. Sketch a smooth oval through all four points.

The four points you plot in steps 3–4 are the vertices (endpoints of the major axis) and co-vertices (endpoints of the minor axis). Which pair is which depends on whether $a$ or $b$ is larger.

Ellipse Features vs. Equation Components

The values of $a$ and $b$ control more than just size:

• Major axis length = $2 \times$ (the larger of $a$ or $b$)
• Minor axis length = $2 \times$ (the smaller of $a$ or $b$)
• Orientation: If $a > b$, the major axis is horizontal. If $b > a$, the major axis is vertical.

Finding the foci: The foci always lie on the major axis, each a distance $c$ from the center, where:

$c^2 = (\text{larger denominator}) - (\text{smaller denominator})$

More specifically:

• Horizontal major axis ($a > b$): $c^2 = a^2 - b^2$, foci at $(h \pm c,\; k)$
• Vertical major axis ($b > a$): $c^2 = b^2 - a^2$, foci at $(h,\; k \pm c)$

Eccentricity measures how "stretched" the ellipse is: $e = \frac{c}{(\text{semi-major axis length})}$. An eccentricity near 0 means the ellipse is nearly circular; closer to 1 means it's very elongated.

Area of an ellipse: $A = \pi ab$

Real-World Applications of Ellipses

• Planetary orbits follow elliptical paths with the Sun at one focus (Kepler's First Law). Earth's orbit, for example, has an eccentricity of about 0.017, so it's very close to circular.
• Whispering galleries are elliptical rooms (like the one in St. Paul's Cathedral) where a whisper at one focus reflects off the walls and converges at the other focus, making it clearly audible across the room.
• Architecture and engineering use elliptical arches and gears. The Colosseum in Rome has an elliptical footprint.

Solving applied problems typically requires you to:

1. Extract known information (foci locations, axis lengths, specific points on the ellipse).
2. Use that information to find $a$, $b$, and the center.
3. Write the standard form equation, then answer whatever the problem asks (coordinates of foci, area, etc.).

Additional Ellipse Properties

• An ellipse is a conic section, formed when a plane slices through a cone at an angle that isn't steep enough to produce a parabola or hyperbola.
• The defining property ties back to the foci: for any point $P$ on the ellipse, the sum of distances $PF_1 + PF_2 = 2a$ (where $a$ is the semi-major axis length). This constant-sum property is what gives the ellipse its shape.
• Each ellipse has two directrices, lines perpendicular to the major axis. The ratio of a point's distance to a focus versus its distance to the corresponding directrix equals the eccentricity $e$.
• The latus rectum is a chord through a focus, perpendicular to the major axis. Its length is $\frac{2b^2}{a}$ (when $a$ is the semi-major axis). This value is useful for understanding how "wide" the ellipse is near each focus.
• The auxiliary circle has the same center as the ellipse and a radius equal to the semi-major axis. It's a helpful construction for parametric representations of the ellipse.