Hyperbolas Section 10.4

1 st Definiton

A hyperbola is a conic section formed when a plane intersects both cones.

2 nd Definition

A hyperbola is the set of all points (x, y) in the plane the difference of whose distances from two distinct fixed point (foci) is a positive constant. d1d1 d2d2 |d 2 – d 1 | = constant Turn on the N- Spire Calculator. Open the file Hyperbola Construction.

The graph of a hyperbola has two disconnected branches. The line through the two foci intersects the hyperbola at its two vertices. The line segment connecting the vertices is the transverse axis, and the midpoint of the transverse axis is the center of the hyperbola.

The vertices are a units from the center, and the foci are c units from the center. c a

General Equation of a Hyperbola Ax 2 + Cy 2 + Dx + Ey + F = 0 Either A or C will be negative.

Standard Equation of a Hyperbola

The standard form of the equation of a hyperbola with center (h, k) is Transverse axis is horizontal Transverse axis is vertical c 2 = a 2 + b 2

Each hyperbola has two asymptotes that intersect at the center of the hyperbola. The asymptotes pass through the vertices of a rectangle of dimensions 2a by 2b, with its center at (h, k). The line segment of length 2b joining (h, k + b) and (h, k – b) or (h + b, k) and (h – b, k) is the conjugate axis of the hyperbola.

Asymptotes of a Hyperbola

The equations of the asymptotes of a hyperbola are Transverse axis is horizontal Transverse axis is vertical Is the eccentricity greater or less than 1? Why?

Examples Write each hyperbola in standard form then find center, vertices, foci, length of the transverse and conjugate axes, eccentricity, and the equations of the asymptotes. Graph the hyperbola.

1.

center: vertices: transverse axis: conjugate axis: foci: eccentricity: (1, 4) (-3, 4), (5, 4) 2a = 8 2b = 2 asymptotes:

V1V1 V2V2

center: vertices: transverse axis: conjugate axis: foci: eccentricity: (1, -3) (1, -5), (1, -1) 2a = 4 2b = 6 asymptotes:

Examples Given the information, write the hyperbola in standard form.

1. center (0, 0); a = 4, b = 2; horizontal transverse axis

2. center (-2, 1); a = 5, c = 8; vertical transverse axis

3.vertices at (-5, 1) and (-5, 7); conjugate axis length of 12 units center: (-5, 4) vertical transverse axis 2a = 6 so a = 3 2b = 12 so b = 6 END OF 1 ST DAY

If we have a hyperbola with a transverse axis distance of a and an eccentricity of e, each directrix of the hyperbola is defined as the line perpendicular to the line containing the transverse axis at a distance from the center of Therefore each directrix does not intersect the hyperbola.

Focus-Directrix Property of Hyperbola This property explains how the directrix relates to a hyperbola. THIS IS THE THIRD DEFINITION OF A HYPERBOLA.

A hyperbola is the set of all points P such that the distance from a point on the hyperbola to the focus F is e times the distance from the same point to the associated directrix. PF 1 = e AP P F1F1 Directrix F2F2 A

Example Graph the hyperbola showing the center, vertices, foci, the asymptotes, and each directrix. Find eccentricity and the equations of the directrix. 16y 2 − 9x 2 = 144

eccentricity: directrix:

Example Find the center, vertices, foci, eccentricity, and equations of asymptotes and the directrix. 9x 2 − 4y 2 − 90x −24y = −153

center: vertices: foci: eccentricity: equations of asymptotes: equation of each directrix: (5, -3) (7, -3), (3, -3)

Example Given a hyperbola with a focus at (5, 0), an associated directrix at, and a point on the hyperbola at (3, 0), find the eccentricity.

PF =2 AP = PF = e AP END OF 2 nd DAY

Day 3

State the general equation of conic sections. Ax 2 + Cy 2 + Dx + Ey + F = 0 A conic section is a(n) 1.circle if 2.parabola if 3.ellipse if 4.hyperbola if A = C AC = 0 AC > 0 AC < 0

Example Classify each of the following. Explain your answers. 1.4x 2 + 5y 2 − 9x + 8y = 0 ellipse because AC = x 2 − 5x + 7y − 8 = 0 parabola because AC = 0 3.7x 2 + 7y 2 − 9x + 8y − 16 = 0 circle because A = C

4.4x 2 − 5y 2 − x + 8y + 1 = 0 hyperbola because AC = -20

Example A passageway in a house is to have straight sides and a semielliptically-arched top. The straight sides are 5 feet tall and the passageway is 7 feet tall at its center and 6 feet wide. Where should the foci be located to make the template for the arch?

5 feet 6 feet 7 feet 2a = 6 a = 3 b = 7 − 5 = 2 The foci should be about feet right and left of the center of the semiellipse.

Example There is a listening station located at A(2200, 0) (in feet) and another at B(-2200, 0). An explosion is heard at station A one second before it is heard at station B. Where was the explosion located? Sound travels at 1100 feet per second. The listening stations can be considered foci of a hyperbola. So c = 2200 and the center is (0, 0). The explosion occurred on the right branch of this hyperbola since station B heard it one second after station A.

d2d2 d1d1 d 2 − d 1 = 1100 d 2 − d 1 = 2a 2a = 1100 a = 550 The difference is equal to 1100 because station A heard the explosion 1 second before station B.

The location of the explosion is the equation of this hyperbola.