7.5 Conic Sections

Conic Sections

Conic sections are the curves you get by slicing a double cone at different angles: parabolas, ellipses, and hyperbolas. In Calculus II, these curves matter because they have elegant polar representations and show up constantly in physics (orbital mechanics, optics, signal processing). This section covers their equations in both standard and polar form, eccentricity, classification, and how to sketch them.

Conic Sections

Equations of Conic Sections

Each conic is defined by a geometric relationship between points and fixed references (foci, directrices). That geometric definition is what gives rise to the standard equation.

Parabolas

A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).

• Vertical axis: $y = a(x - h)^2 + k$
• Horizontal axis: $x = a(y - k)^2 + h$
• The vertex is at $(h, k)$
• The sign of $a$ controls direction: positive $a$ opens upward (or rightward), negative $a$ opens downward (or leftward)
• The distance from vertex to focus is $\frac{1}{4|a|}$

Applications: projectile trajectories, satellite dish reflectors, car headlight mirrors. In each case, the reflective property of parabolas (all parallel rays reflect through the focus) is what matters.

Ellipses

An ellipse is the set of all points where the sum of distances to two fixed points (the foci) is constant.

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

• Center at $(h, k)$
• Semi-major axis $a$ (the longer one) and semi-minor axis $b$ (the shorter one), with $a > b$
• If the larger denominator is under the $x$-term, the major axis is horizontal; if it's under the $y$-term, the major axis is vertical
• Foci lie along the major axis at distance $c = \sqrt{a^2 - b^2}$ from the center

Applications: planetary orbits (Kepler's first law says planets orbit in ellipses with the Sun at one focus), whispering galleries (St. Paul's Cathedral), elliptical gears.

Hyperbolas

A hyperbola is the set of all points where the difference of distances to two foci is constant.

• Horizontal transverse axis: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$
• Vertical transverse axis: $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$
• Center at $(h, k)$
• $a$ is the semi-transverse axis (distance from center to each vertex), $b$ is the semi-conjugate axis
• Foci are at distance $c = \sqrt{a^2 + b^2}$ from the center
• Asymptotes have slopes $\pm \frac{b}{a}$ (for horizontal transverse axis)

Applications: sonic boom shock waves, LORAN navigation (which uses time differences of signals from two stations), cooling tower structures.

Eccentricity in Conic Analysis

Eccentricity ($e$) is a single number that tells you how "stretched" a conic is relative to a circle. It's what unifies all conics into one family:

• Circle: $e = 0$
• Ellipse: $0 < e < 1$
• Parabola: $e = 1$
• Hyperbola: $e > 1$

For an ellipse: $e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}}$, where $a > b$. As $e$ approaches 1, the ellipse gets more elongated. Earth's orbit has $e \approx 0.017$ (nearly circular), while Halley's Comet has $e \approx 0.967$ (extremely elongated).

For a hyperbola: $e = \frac{c}{a} = \sqrt{1 + \frac{b^2}{a^2}}$. Larger $e$ means the branches open more widely.

For a parabola: $e = 1$ always. There's no variation.

Standard vs. Polar Form Equations

This is where conic sections connect to the polar coordinates theme of this unit. All three conics can be written with a single polar equation when one focus is placed at the origin.

Polar form:

$r = \frac{ed}{1 + e\cos\theta}$

or variants with $-\cos\theta$, $\pm\sin\theta$ depending on the orientation and which directrix you reference.

• $e$ is the eccentricity
• $d$ is the distance from the focus to the corresponding directrix (sometimes written as $p$ for the semi-latus rectum, where the semi-latus rectum $\ell = ed$)

The type of conic depends entirely on $e$: plug in $e < 1$ and you get an ellipse, $e = 1$ gives a parabola, $e > 1$ gives a hyperbola. That's the power of this form.

Relating polar parameters to standard form:

• For an ellipse, the semi-latus rectum is $\ell = \frac{b^2}{a}$
• For a hyperbola, the semi-latus rectum is also $\ell = \frac{b^2}{a}$
• For a parabola $y = ax^2$ (vertex at origin), the focal distance is $\frac{1}{4a}$, and the semi-latus rectum equals twice the focal distance

Classification of Second-Degree Equations

Any conic can be written in the general second-degree form:

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

You classify it using the discriminant $\Delta = B^2 - 4AC$:

• $\Delta < 0$: ellipse (or circle, or degenerate)
• $\Delta = 0$: parabola (or degenerate)
• $\Delta > 0$: hyperbola (or degenerate)

Special cases to watch for:

• Circle: $A = C$ and $B = 0$ (equal coefficients on $x^2$ and $y^2$, no cross term)
• Degenerate conics: the equation might factor into lines, a single point, or have no real solutions. Always check after classifying.

If $B \neq 0$, the conic's axes are rotated relative to the coordinate axes. You can eliminate the $xy$ term by rotating axes through angle $\theta$ where $\cot 2\theta = \frac{A - C}{B}$.

Sketching Conics from Equations

Parabolas:

1. Rewrite in standard form to identify the vertex $(h, k)$
2. Find the focus: it's $\frac{1}{4|a|}$ units from the vertex along the axis of symmetry
3. Find the directrix: same distance from the vertex, opposite side
4. Plot the vertex, draw the axis of symmetry, and sketch the curve opening toward the focus

Ellipses:

1. Rewrite in standard form to identify the center $(h, k)$

2. Read off $a$ and $b$ from the denominators ($a$ is the larger value)

3. Plot the four endpoints of the axes: vertices at $\pm a$ along the major axis, co-vertices at $\pm b$ along the minor axis

4. Find foci at $c = \sqrt{a^2 - b^2}$ from center along the major axis

5. Sketch a smooth oval through the four endpoints

Hyperbolas:

1. Rewrite in standard form to identify the center $(h, k)$ and which variable is positive
2. Plot vertices at $\pm a$ from center along the transverse axis
3. Find foci at $c = \sqrt{a^2 + b^2}$ from center along the transverse axis
4. Draw the asymptotes: lines through the center with slopes $\pm \frac{b}{a}$ (horizontal case) or $\pm \frac{a}{b}$ (vertical case)
5. Sketch each branch approaching but never touching the asymptotes

Applications in Physics and Engineering

• Parabolas and reflection: Any ray parallel to the axis of a parabola reflects through the focus. This is why satellite dishes, radio telescopes, and headlights all use parabolic shapes.
• Ellipses and orbits: Kepler's first law states that planets travel in elliptical orbits with the Sun at one focus. The eccentricity determines how circular or elongated the orbit is.
• Hyperbolas and navigation: The LORAN system determines position by measuring the time difference of radio signals from two stations. Each constant time difference traces out a hyperbola, and the intersection of two hyperbolas gives your location.
• Cooling towers: Many power plant cooling towers have a hyperboloid shape (a hyperbola rotated around an axis), which provides structural strength with minimal material.

Advanced Conic Section Concepts

Rotation of axes is the main advanced technique here. When a general second-degree equation has a nonzero $Bxy$ term, the conic is tilted. To analyze it:

1. Compute the rotation angle: $\cot 2\theta = \frac{A - C}{B}$

2. Apply the substitution $x = X\cos\theta - Y\sin\theta$, $y = X\sin\theta + Y\cos\theta$

3. The resulting equation in $X, Y$ will have no cross term, so you can classify and sketch it using standard methods

Locus definition: Each conic is defined as a locus (set of all points satisfying a distance condition). This geometric definition is what connects the algebraic equations to the physical properties. For instance, the reflective property of parabolas follows directly from the equidistant-from-focus-and-directrix definition.