Pre-Calculus - Week 8

Conic Sections: Ellipses and Hyperbolas

Ellipses and hyperbolas complete the family of conic sections. Ellipses model planetary orbits (Kepler's first law), while hyperbolas appear in navigation systems, sonic booms, and the shapes of mirrors in telescopes. Both curves are defined by their relationship to two foci, a beautiful geometric property with deep physical meaning.

Learning Objectives

- Write the standard form of an ellipse and identify center, vertices, and foci. - Graph an ellipse given its equation. - Write the standard form of a hyperbola and identify center, vertices, foci, and asymptotes. - Graph a hyperbola including its asymptotes. - Complete the square to convert general conic form to standard form. - Identify which conic section is represented by a general second-degree equation. - Apply the reflective property of ellipses and hyperbolas in context.

1. The Ellipse: Definition

An ellipse is the set of all points where the sum of distances from two fixed foci is constant. If the foci are F1 and F2, and the constant sum is 2a, then for any point P on the ellipse: dist(P, F1) + dist(P, F2) = 2a. Key terms: - a: semi-major axis (half the longer axis) - b: semi-minor axis (half the shorter axis) - c: distance from center to each focus - Relationship: a^2 = b^2 + c^2 (always: a > b > 0, c < a)

2. Standard Form of an Ellipse

Center at (h, k): Horizontal major axis: (x-h)^2/a^2 + (y-k)^2/b^2 = 1 (a > b) Vertical major axis: (x-h)^2/b^2 + (y-k)^2/a^2 = 1 (a > b) The major axis has length 2a (longer). The minor axis has length 2b (shorter). Vertices are at the ends of the major axis. Co-vertices are at the ends of the minor axis. Foci are inside the ellipse, at distance c from center along the major axis. Example: x^2/25 + y^2/9 = 1. a^2=25, b^2=9 (horizontal since 25>9). a=5, b=3. c=sqrt(25-9)=4. Center (0,0). Vertices: (±5,0). Co-vertices: (0,±3). Foci: (±4,0).

3. Ellipse with Center Not at Origin

Example: (x-2)^2/16 + (y+1)^2/4 = 1. a^2=16, b^2=4. a=4, b=2. Horizontal major axis. c = sqrt(16-4) = sqrt(12) = 2*sqrt(3). Center: (2, -1). Vertices: (2±4, -1) = (6, -1) and (-2, -1). Co-vertices: (2, -1±2) = (2, 1) and (2, -3). Foci: (2 ± 2*sqrt(3), -1).

4. Writing Ellipse Equations

Example 1: Write the equation of the ellipse with center (0,0), vertices (±6,0), and co-vertices (0,±4). a=6, b=4. x^2/36 + y^2/16 = 1. Example 2: Write the equation with foci (0,±3) and vertices (0,±5). Vertical major axis. a=5, c=3. b^2=25-9=16. x^2/16 + y^2/25 = 1.

5. The Hyperbola: Definition

A hyperbola is the set of all points where the DIFFERENCE of distances from two foci is constant. |dist(P,F1) - dist(P,F2)| = 2a. A hyperbola has TWO branches. Key relationship (note the MINUS sign, opposite of ellipse): c^2 = a^2 + b^2 (here c > a, unlike the ellipse). Every hyperbola has two asymptotes — lines the branches approach but never touch.

6. Standard Form of a Hyperbola

Horizontal (opens left/right): (x-h)^2/a^2 - (y-k)^2/b^2 = 1. Asymptotes: y - k = ±(b/a)(x - h). Vertices: (h±a, k). Vertical (opens up/down): (y-k)^2/a^2 - (x-h)^2/b^2 = 1. Asymptotes: y - k = ±(a/b)(x - h). Vertices: (h, k±a). Example: x^2/9 - y^2/16 = 1. Horizontal. a=3, b=4. Center (0,0). c=sqrt(9+16)=5. Vertices: (±3,0). Foci: (±5,0). Asymptotes: y=±(4/3)x.

7. Graphing a Hyperbola

Steps: 1. Find center, a, b, c. 2. Draw the central rectangle: width 2a (horizontal) or 2b (vertical), height 2b or 2a. 3. Draw the asymptotes as the diagonals of this rectangle. 4. Plot vertices on the appropriate axis. 5. Sketch both branches curving away from center through vertices, approaching the asymptotes. Example: y^2/4 - x^2/9 = 1. Vertical. a=2, b=3. Center (0,0). Vertices (0,±2). Asymptotes y=±(2/3)x. c=sqrt(4+9)=sqrt(13). Foci (0,±sqrt(13)).

8. Identifying Conics from General Form

The general second-degree equation: Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. When B = 0 (no xy term): - A = C (and same sign): circle. - A and C same sign, A ≠ C: ellipse. - A and C opposite signs: hyperbola. - A = 0 or C = 0 (not both): parabola. Example 1: 4x^2 + 9y^2 - 36 = 0. Rewrite: x^2/9 + y^2/4 = 1. Ellipse. Example 2: x^2 - 4y^2 = 16. Rewrite: x^2/16 - y^2/4 = 1. Hyperbola. Example 3: x^2 + y^2 - 6x = 7. Circle. Example 4: x^2 - 4x - y = 0. Parabola (only one squared term).

9. Completing the Square for Conics

Convert 4x^2 + 9y^2 - 16x + 36y + 16 = 0 to standard form. Group: (4x^2 - 16x) + (9y^2 + 36y) = -16. Factor: 4(x^2 - 4x) + 9(y^2 + 4y) = -16. Complete square: 4(x-2)^2 - 16 + 9(y+2)^2 - 36 = -16. 4(x-2)^2 + 9(y+2)^2 = 36. Divide by 36: (x-2)^2/9 + (y+2)^2/4 = 1. Ellipse, center (2,-2), a=3, b=2.

10. Applications

Planetary orbits (Kepler): planets travel in ellipses with the sun at one focus. Whispering galleries: elliptical rooms where sound from one focus travels to the other — the dome of the U.S. Capitol uses this effect. Navigation (LORAN): hyperbolas can locate a ship's position from the time difference in radio signals received from two stations. Telescopes: Cassegrain telescopes use a parabolic primary mirror and a hyperbolic secondary mirror — the focus of the hyperbola coincides with the focus of the parabola.

11. Common Mistakes

- Ellipse: forgetting a^2 > b^2 and using the wrong denominator for the major axis. - Confusing the foci formula for ellipses (c^2 = a^2 - b^2) with hyperbolas (c^2 = a^2 + b^2). - For hyperbolas: the 'a' goes under the POSITIVE term (not necessarily the x term). - Asymptotes for x^2/a^2 - y^2/b^2 = 1 are y = ±(b/a)x, not ±(a/b)x. - Forgetting that hyperbolas have two branches; only sketching one.