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Lecture Notes on Hyperbolas

Jun 26, 2024

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Lecture on Hyperbolas

Introduction to Hyperbolas

Focus on horizontally and vertically oriented hyperbolas
Formulas for hyperbolas centered at the origin
- Horizontal:
  $x^2 / a^2 - y^2 / b^2 = 1$
- Vertical:
  $y^2 / a^2 - x^2 / b^2 = 1$

Vertices and Foci

For horizontal hyperbola: Vertices are
$\pm a, 0$
, Foci are
$\pm c, 0$
For vertical hyperbola: Vertices are
$0, \pm a$
, Foci are
$0, \pm c$
Relation:
$c^2 = a^2 + b^2$

Differences from Ellipses

For ellipses:
$c^2 = a^2 - b^2$
Key Point: For ellipses,
$a > b$
, but for hyperbolas, a and b do not have a fixed size relation.

Transverse Axis

Connects the two vertices
Length:
$2a$
Horizontal for horizontal hyperbolas, vertical for vertical hyperbolas
Distance between foci:
$2c$

Asymptotes

Horizontal hyperbola:
$y = \pm \frac{b}{a}x$
Vertical hyperbola:
$y = \pm \frac{a}{b}x$

Example Problem 1

Identify center, vertices, foci, and asymptotes for given hyperbola

Given Hyperbola

Equation:
$\frac{x^2}{4} - \frac{y^2}{9} = 1$
Center: (0, 0)
$\a = 2$
$\b = 3$
Vertices: (±2, 0)
Foci: (±√13, 0)
Asymptotes:
$y = \pm \frac{3}{2}x$

Graphing the Hyperbola

Center at the origin
Travel a units right and left, b units up and down
Draw rectangle, then asymptotes through diagonals
Openings along x-axis

Standard Form Adjustments

Horizontal at Center not at Origin

Equation:
$\frac{x - h^2}{a^2} - \frac{y - k^2}{b^2} = 1$
Vertices:
$\h \pm a, k$
Foci:
$\h \pm c, k$
Asymptotes:
$y - k = \pm \frac{b}{a}(x - h)$

Vertical at Center not at Origin

Equation:
$\frac{y - k^2}{a^2} - \frac{x - h^2}{b^2} = 1$
Vertices:
$h, k \pm a$
Foci:
$h, k \pm c$
Asymptotes:
$y - k = \pm \frac{a}{b}(x - h)$

Example Problem 2

Convert given hyperbola to standard form

Given Hyperbola

$144 \frac{y^2}{9} - 144 \frac{x^2}{16} = 144$
Convert to:
$\frac{y^2}{16} - \frac{x^2}{9}if= 1$
Center at origin
![equation](https://latex.codecogs.com/png.latex%736cn- Vertices:
$(0, \pm 4)$
Foci: (0, ±5)
Asymptotes: y = ±4/3x

Adjustments when Center is Moved

Horizontal Hyperbola

$y - k = \pm \frac{b}{a}(x - h)$
Vertices:
$\h \pm a, k$
Foci:
$\h \pm c, k$

Vertical Hyperbola

Equation:
$\frac{y - k^2}{a^2} - \frac{x - h^2}{b^2} = 1$
Vertices:
$h, k \pm a$
Foci:
$h, k \pm c$

Problem 3

Details for Hyperbola Not Centered at Origin

Center: (3, -2)
$\a=2, b=3, c= \sqrt{13}$
Vertices: (3±2, -2) or (1, -2) and (5, -2)
Foci: (3±√13, -2) or (3±3.6, -2) approximately (6.6, -2) and (0.4, -2)
Asymptote Equation:
$y = \pm \frac{3}{2}(x-3)-2, y+2=\pm\frac{3}{2}(x-3)$

Final Problem

Given Equation

$\frac{y+1}{9} - \frac{x-2}{16}=1$
Center: (2, 1),
$a= 3, b=4, c=5$
Vertices: (2, 1±3) or (2, 4) and (2, -2)
Foci: (2, 1±5) or (2, 6) and (2, -4)
Asymptotes: y ± 1
$=\pm\frac{a}{b}(x-2)$
Domain: All real values
Range: Interval notation:
$y\in(-\infty,-2)\cup(4,\infty)$

Conclusion

Difference in hyperbola and ellipse equations, vertex and focus definitions
Importance of transverse axis and asymptotes
Remember to convert equations and locate center, vertices, and foci correctly

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