10.2 Ellipses and Hyperbolas

Ellipses and hyperbolas are two conic sections defined by their relationship to two fixed points called foci. An ellipse involves a constant sum of distances to the foci, while a hyperbola involves a constant difference. That single distinction drives all the differences in their equations, shapes, and graphs.

This guide covers the standard equations for both curves, how to graph them, how to derive equations from given information, and where these shapes show up in the real world.

Ellipses and Hyperbolas: Key Features

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Ellipses

An ellipse is the set of all points where the sum of the distances to two fixed points (the foci) is constant. Geometrically, it's the closed oval you get when a plane slices through a cone at an angle.

• Standard form (center at the origin): $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
  • $a$ = semi-major axis length (the longer one), $b$ = semi-minor axis length
  • If $a^2$ is under $x^2$, the major axis is horizontal. If $a^2$ is under $y^2$, the major axis is vertical.
• Foci lie on the major axis at $(\pm c, 0)$ or $(0, \pm c)$, where $c^2 = a^2 - b^2$
  • Since $c^2 = a^2 - b^2$, the foci are always inside the ellipse ($c < a$).
• Eccentricity: $e = \frac{c}{a}$, where $0 < e < 1$
  • Values close to 0 produce a nearly circular ellipse. Values close to 1 produce a long, narrow one.

Hyperbolas

A hyperbola is the set of all points where the absolute difference of the distances to two fixed points (the foci) is constant. It consists of two separate branches that open away from each other.

• Standard form (center at the origin):
  • Horizontal transverse axis: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
  • Vertical transverse axis: $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$
  • $a$ = semi-transverse axis length, $b$ = semi-conjugate axis length
  • Quick way to tell orientation: whichever variable is positive tells you the direction the branches open.
• Foci lie on the transverse axis at $(\pm c, 0)$ or $(0, \pm c)$, where $c^2 = a^2 + b^2$
  • Notice the plus sign here, compared to the minus sign for ellipses. This means $c > a$, so the foci sit outside the vertices.
• Eccentricity: $e = \frac{c}{a}$, where $e > 1$
  • Values close to 1 mean the branches are narrow and steep. Larger values mean the branches open wider.
• Asymptotes are lines the branches approach but never cross:
  • Horizontal transverse axis: $y = \pm\frac{b}{a}x$
  • Vertical transverse axis: $y = \pm\frac{a}{b}x$

Graphing Ellipses and Hyperbolas

Graphing Ellipses

The general form equation of an ellipse with center $(h, k)$ is:

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

To graph it:

1. Identify the center $(h, k)$ and plot it.
2. Determine which axis is major. The larger denominator corresponds to the major axis. If the larger denominator is under $(x-h)^2$, the major axis is horizontal; if it's under $(y-k)^2$, it's vertical.
3. Plot the vertices. Move $a$ units along the major axis from the center in both directions. These are the vertices.
4. Plot the co-vertices. Move $b$ units along the minor axis from the center in both directions.
5. Sketch the curve as a smooth oval through all four points.

Example: For $\frac{(x-1)^2}{16} + \frac{(y+2)^2}{9} = 1$, the center is $(1, -2)$, $a = 4$ (horizontal), $b = 3$ (vertical). Vertices at $(-3, -2)$ and $(5, -2)$; co-vertices at $(1, 1)$ and $(1, -5)$.

Graphing Hyperbolas

The general form equations of a hyperbola with center $(h, k)$ are:

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

To graph it:

1. Identify the center $(h, k)$ and the orientation (which variable's fraction is positive).
2. Plot the vertices. Move $a$ units from the center along the transverse axis in both directions.
3. Draw the central rectangle. From the center, go $a$ units along the transverse axis and $b$ units along the conjugate axis in each direction. This rectangle won't appear on the final graph, but it guides your asymptotes.
4. Draw the asymptotes as diagonal lines through opposite corners of that rectangle.
5. Sketch each branch starting from a vertex, curving outward and approaching (but never touching) the asymptotes.

Converting to Standard Form

If you're given an equation like $4x^2 - 9y^2 + 16x + 18y - 29 = 0$, you'll need to complete the square:

1. Group the $x$ terms and $y$ terms together. Factor out the leading coefficients from each group.
2. Complete the square for both $x$ and $y$.
3. Divide both sides so the right side equals 1.

This process works identically for both ellipses and hyperbolas.

Equations of Ellipses and Hyperbolas

Finding Equations of Ellipses

Given center, vertices, and co-vertices:

1. Find $a$ (distance from center to a vertex) and $b$ (distance from center to a co-vertex).
2. Plug $h$, $k$, $a$, and $b$ into the standard form equation. Make sure $a^2$ goes under the variable that matches the direction of the major axis.

Given foci and a point on the ellipse:

1. The sum of distances from any point on the ellipse to the two foci equals $2a$. Calculate this sum using the given point.

2. Use the foci coordinates to find $c$ (distance from center to a focus).

3. Solve $b^2 = a^2 - c^2$ to get $b$.

4. Write the equation in standard form.

Finding Equations of Hyperbolas

Given center, vertices, and a point on the hyperbola:

1. Find $a$ from the distance between the center and a vertex.
2. Substitute the coordinates of the given point into the standard form equation and solve for $b$.
3. Write the full equation.

Given foci and a point on the hyperbola:

1. The absolute difference of distances from any point on the hyperbola to the two foci equals $2a$. Calculate this using the given point.

2. Use the foci to find $c$.

3. Solve $b^2 = c^2 - a^2$ to get $b$.

4. Write the equation in standard form.

Given asymptotes and a point on the hyperbola:

1. Find the center by solving for the intersection of the two asymptotes.
2. The slopes of the asymptotes give you the ratio $\frac{b}{a}$ (horizontal) or $\frac{a}{b}$ (vertical). Express $b$ in terms of $a$.
3. Substitute the given point into the standard form equation to solve for $a$ and $b$ individually.
4. Write the full equation.

Applications of Ellipses and Hyperbolas

Ellipses in Real-World Scenarios

Planetary orbits: Kepler's first law states that planets orbit the Sun in ellipses with the Sun at one focus. Earth's orbit has a very low eccentricity ($e \approx 0.017$), so it's nearly circular. Mercury's orbit is more eccentric ($e \approx 0.206$), making it noticeably elongated. Halley's Comet has $e \approx 0.967$, producing an extremely stretched ellipse.

Architecture and engineering: Elliptical shapes appear in arches and domes because they distribute loads efficiently. The Oval Office in the White House and the U.S. Capitol dome both use elliptical geometry.

Optics (the reflective property): An ellipse has the property that any signal emitted from one focus reflects off the surface and converges at the other focus. This principle is used in whispering galleries (like the one in the U.S. Capitol) and in medical devices like lithotripters, which use elliptical reflectors to focus shock waves on kidney stones. Telescope mirrors also use elliptical curvature to focus light.

Hyperbolas in Real-World Scenarios

Sonic booms and shock waves: When an object moves faster than the speed of sound, it creates a conical shock wave. A cross-section of that cone is a hyperbola. The Mach number (ratio of the object's speed to the speed of sound) determines the angle between the asymptotes of that hyperbola.

Architecture: Hyperbolic shapes appear in cooling tower designs at power plants. The curved profile promotes natural air circulation and provides structural strength with minimal material. The Gateway Arch in St. Louis follows an inverted weighted catenary, which is closely related to a hyperbolic cosine curve.

Optics and navigation: Hyperbolic mirrors are used in some telescope designs (like Cassegrain reflectors) and in certain headlight reflectors to direct light. Historically, the LORAN navigation system located ships by measuring time differences of radio signals from fixed stations, which traced out hyperbolas on a map.