Contents

Preface
1Prerequisites
2Equations and Inequalities
3Functions
4Linear Functions
5Polynomial and Rational Functions
6Exponential and Logarithmic Functions
7The Unit Circle: Sine and Cosine Functions
8Periodic Functions
9Trigonometric Identities and Equations
10Further Applications of Trigonometry
11Systems of Equations and Inequalities
12Analytic Geometry
13Sequences, Probability, and Counting Theory
AProofs, Identities, and Toolkit Functions
Answer Key
Index

Chapter 12

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$x^{2} + \frac{y^{2}}{16} = 1 x^{2} + \frac{y^{2}}{16} = 1$

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$\frac{{(x - 1)}^{2}}{16} + \frac{{(y - 3)}^{2}}{4} = 1 \frac{{(x - 1)}^{2}}{16} + \frac{{(y - 3)}^{2}}{4} = 1$

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center: $(0, 0); (0, 0);$ vertices: $(\pm 6, 0); (\pm 6, 0);$ co-vertices: $(0, \pm 2); (0, \pm 2);$ foci: $(\pm 4 \sqrt{2}, 0) (\pm 4 \sqrt{2}, 0)$

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Standard form: $\frac{x^{2}}{16} + \frac{y^{2}}{49} = 1; \frac{x^{2}}{16} + \frac{y^{2}}{49} = 1;$ center: $(0, 0); (0, 0);$ vertices: $(0, \pm 7); (0, \pm 7);$ co-vertices: $(\pm 4, 0); (\pm 4, 0);$ foci: $(0, \pm \sqrt{33}) (0, \pm \sqrt{33})$

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Center: $(4, 2); (4, 2);$ vertices: $(- 2, 2) (- 2, 2)$ and $(10, 2); (10, 2);$ co-vertices: $(4, 2 - 2 \sqrt{5}) (4, 2 - 2 \sqrt{5})$ and $(4, 2 + 2 \sqrt{5}); (4, 2 + 2 \sqrt{5});$ foci: $(0, 2) (0, 2)$ and $(8, 2) (8, 2)$

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$\frac{{(x - 3)}^{2}}{4} + \frac{{(y + 1)}^{2}}{16} = 1; \frac{{(x - 3)}^{2}}{4} + \frac{{(y + 1)}^{2}}{16} = 1;$ center: $(3, - 1); (3, - 1);$ vertices: $(3, - 5) (3, - 5)$ and $(3, 3); (3, 3);$ co-vertices: $(1, - 1) (1, - 1)$ and $(5, - 1); (5, - 1);$ foci: $(3, - 1 - 2 \sqrt{3}) (3, - 1 - 2 \sqrt{3})$ and $(3, - 1+ 2 \sqrt{3}) (3, - 1+ 2 \sqrt{3})$

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ⓐ $\frac{x^{2}}{57, 600} + \frac{y^{2}}{25, 600} = 1 \frac{x^{2}}{57, 600} + \frac{y^{2}}{25, 600} = 1$
ⓑ The people are standing 358 feet apart.

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Vertices: $(\pm 3, 0); (\pm 3, 0);$ Foci: $(\pm \sqrt{34}, 0) (\pm \sqrt{34}, 0)$

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$\frac{y^{2}}{4} - \frac{x^{2}}{16} = 1 \frac{y^{2}}{4} - \frac{x^{2}}{16} = 1$

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$\frac{{(y - 3)}^{2}}{25} + \frac{{(x - 1)}^{2}}{144} = 1 \frac{{(y - 3)}^{2}}{25} + \frac{{(x - 1)}^{2}}{144} = 1$

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vertices: $(\pm 12, 0); (\pm 12, 0);$ co-vertices: $(0, \pm 9); (0, \pm 9);$ foci: $(\pm 15, 0); (\pm 15, 0);$ asymptotes: $y = \pm \frac{3}{4} x; y = \pm \frac{3}{4} x;$

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center: $(3, - 4); (3, - 4);$ vertices: $(3, - 14) (3, - 14)$ and $(3, 6); (3, 6);$ co-vertices: $(- 5, - 4); (- 5, - 4);$ and $(11, - 4); (11, - 4);$ foci: $(3, - 4 - 2 \sqrt{41}) (3, - 4 - 2 \sqrt{41})$ and $(3, - 4 + 2 \sqrt{41}); (3, - 4 + 2 \sqrt{41});$ asymptotes: $y = \pm \frac{5}{4} (x - 3) - 4 y = \pm \frac{5}{4} (x - 3) - 4$

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The sides of the tower can be modeled by the hyperbolic equation. $\frac{x^{2}}{400} - \frac{y^{2}}{3600} = 1 or \frac{x^{2}}{20^{2}} - \frac{y^{2}}{60^{2}} = 1. \frac{x^{2}}{400} - \frac{y^{2}}{3600} = 1 or \frac{x^{2}}{20^{2}} - \frac{y^{2}}{60^{2}} = 1.$

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Focus: $(- 4, 0); (- 4, 0);$ Directrix: $x = 4; x = 4;$ Endpoints of the latus rectum: $(- 4, \pm 8) (- 4, \pm 8)$

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Focus: $(0, 2); (0, 2);$ Directrix: $y = −2; y = −2;$ Endpoints of the latus rectum: $(\pm 4, 2) . (\pm 4, 2) .$

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$x^{2} = 14 y . x^{2} = 14 y .$

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Vertex: $(8, - 1); (8, - 1);$ Axis of symmetry: $y = −1; y = −1;$ Focus: $(9, - 1); (9, - 1);$ Directrix: $x = 7; x = 7;$ Endpoints of the latus rectum: $(9, - 3) (9, - 3)$ and $(9, 1) . (9, 1) .$

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Vertex: $(- 2, 3); (- 2, 3);$ Axis of symmetry: $x = −2; x = −2;$ Focus: $(- 2, - 2); (- 2, - 2);$ Directrix: $y = 8; y = 8;$ Endpoints of the latus rectum: $(- 12, - 2) (- 12, - 2)$ and $(8, - 2) . (8, - 2) .$

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ⓐ $y^{2} = 1280 x y^{2} = 1280 x$
ⓑ The depth of the cooker is 500 mm

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ⓐ hyperbola
ⓑ ellipse

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$\frac{{x^{'}}^{2}}{4} + \frac{{y^{'}}^{2}}{1} = 1 \frac{{x^{'}}^{2}}{4} + \frac{{y^{'}}^{2}}{1} = 1$

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ⓐ hyperbola
ⓑ ellipse

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ellipse; $e = \frac{1}{3}; x = - 2 e = \frac{1}{3};$

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$r = \frac{1}{1 - \cos θ} r = \frac{1}{1 - \cos θ}$

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$4 - 8 x + 3 x^{2} - y^{2} = 0 4 - 8 x + 3 x^{2} - y^{2} = 0$

12.1 Section Exercises

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An ellipse is the set of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant.

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This special case would be a circle.

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It is symmetric about the x-axis, y-axis, and the origin.

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yes; $\frac{x^{2}}{3^{2}} + \frac{y^{2}}{2^{2}} = 1 \frac{x^{2}}{3^{2}} + \frac{y^{2}}{2^{2}} = 1$

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yes; $\frac{x^{2}}{{(\frac{1}{2})}^{2}} + \frac{y^{2}}{{(\frac{1}{3})}^{2}} = 1 \frac{x^{2}}{{(\frac{1}{2})}^{2}} + \frac{y^{2}}{{(\frac{1}{3})}^{2}} = 1$

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11.

$\frac{x^{2}}{2^{2}} + \frac{y^{2}}{7^{2}} = 1; \frac{x^{2}}{2^{2}} + \frac{y^{2}}{7^{2}} = 1;$ Endpoints of major axis $(0, 7) (0, 7)$ and $(0, - 7) . (0, - 7) .$ Endpoints of minor axis $(2, 0) (2, 0)$ and $(- 2, 0) . (- 2, 0) .$ Foci at $(0, 3 \sqrt{5}), (0, - 3 \sqrt{5}) . (0, 3 \sqrt{5}), (0, - 3 \sqrt{5}) .$

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13.

$\frac{x^{2}}{{(1)}^{2}} + \frac{y^{2}}{{(\frac{1}{3})}^{2}} = 1; \frac{x^{2}}{{(1)}^{2}} + \frac{y^{2}}{{(\frac{1}{3})}^{2}} = 1;$ Endpoints of major axis $(1, 0) (1, 0)$ and $(- 1, 0) . (- 1, 0) .$ Endpoints of minor axis $(0, \frac{1}{3}), (0, - \frac{1}{3}) . (0, \frac{1}{3}), (0, - \frac{1}{3}) .$ Foci at $(\frac{2 \sqrt{2}}{3}, 0), (- \frac{2 \sqrt{2}}{3}, 0) . (\frac{2 \sqrt{2}}{3}, 0), (- \frac{2 \sqrt{2}}{3}, 0) .$

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15.

$\frac{{(x - 2)}^{2}}{7^{2}} + \frac{{(y - 4)}^{2}}{5^{2}} = 1; \frac{{(x - 2)}^{2}}{7^{2}} + \frac{{(y - 4)}^{2}}{5^{2}} = 1;$ Endpoints of major axis $(9, 4), (- 5, 4) . (9, 4), (- 5, 4) .$ Endpoints of minor axis $(2, 9), (2, - 1) . (2, 9), (2, - 1) .$ Foci at $(2 + 2 \sqrt{6}, 4), (2 - 2 \sqrt{6}, 4) . (2 + 2 \sqrt{6}, 4), (2 - 2 \sqrt{6}, 4) .$

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17.

$\frac{{(x + 5)}^{2}}{2^{2}} + \frac{{(y - 7)}^{2}}{3^{2}} = 1; \frac{{(x + 5)}^{2}}{2^{2}} + \frac{{(y - 7)}^{2}}{3^{2}} = 1;$ Endpoints of major axis $(- 5, 10), (- 5, 4) . (- 5, 10), (- 5, 4) .$ Endpoints of minor axis $(- 3, 7), (- 7, 7) . (- 3, 7), (- 7, 7) .$ Foci at $(- 5, 7 + \sqrt{5}), (- 5, 7 - \sqrt{5}) . (- 5, 7 + \sqrt{5}), (- 5, 7 - \sqrt{5}) .$

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19.

$\frac{{(x - 1)}^{2}}{3^{2}} + \frac{{(y - 4)}^{2}}{2^{2}} = 1; \frac{{(x - 1)}^{2}}{3^{2}} + \frac{{(y - 4)}^{2}}{2^{2}} = 1;$ Endpoints of major axis $(4, 4), (- 2, 4) . (4, 4), (- 2, 4) .$ Endpoints of minor axis $(1, 6), (1, 2) . (1, 6), (1, 2) .$ Foci at $(1 + \sqrt{5}, 4), (1 - \sqrt{5}, 4) . (1 + \sqrt{5}, 4), (1 - \sqrt{5}, 4) .$

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21.

$\frac{{(x - 3)}^{2}}{{(3 \sqrt{2})}^{2}} + \frac{{(y - 5)}^{2}}{{(\sqrt{2})}^{2}} = 1; \frac{{(x - 3)}^{2}}{{(3 \sqrt{2})}^{2}} + \frac{{(y - 5)}^{2}}{{(\sqrt{2})}^{2}} = 1;$ Endpoints of major axis $(3 + 3 \sqrt{2}, 5), (3 - 3 \sqrt{2}, 5) . (3 + 3 \sqrt{2}, 5), (3 - 3 \sqrt{2}, 5) .$ Endpoints of minor axis $(3, 5 + \sqrt{2}), (3, 5 - \sqrt{2}) . (3, 5 + \sqrt{2}), (3, 5 - \sqrt{2}) .$ Foci at $(7, 5), (- 1, 5) . (7, 5), (- 1, 5) .$

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23.

$\frac{{(x + 5)}^{2}}{{(5)}^{2}} + \frac{{(y - 2)}^{2}}{{(2)}^{2}} = 1; \frac{{(x + 5)}^{2}}{{(5)}^{2}} + \frac{{(y - 2)}^{2}}{{(2)}^{2}} = 1;$ Endpoints of major axis $(0, 2), (- 10, 2) . (0, 2), (- 10, 2) .$ Endpoints of minor axis $(- 5, 4), (- 5, 0) . (- 5, 4), (- 5, 0) .$ Foci at $(- 5 + \sqrt{21}, 2), (- 5 - \sqrt{21}, 2) . (- 5 + \sqrt{21}, 2), (- 5 - \sqrt{21}, 2) .$

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25.

$\frac{{(x + 3)}^{2}}{{(5)}^{2}} + \frac{{(y + 4)}^{2}}{{(2)}^{2}} = 1; \frac{{(x + 3)}^{2}}{{(5)}^{2}} + \frac{{(y + 4)}^{2}}{{(2)}^{2}} = 1;$ Endpoints of major axis $(2, - 4), (- 8, - 4) . (2, - 4), (- 8, - 4) .$ Endpoints of minor axis $(- 3, - 2), (- 3, - 6) . (- 3, - 2), (- 3, - 6) .$ Foci at $(- 3 + \sqrt{21}, - 4), (- 3 - \sqrt{21}, - 4) . (- 3 + \sqrt{21}, - 4), (- 3 - \sqrt{21}, - 4) .$

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27.

Foci $(- 3, - 1 + \sqrt{11}), (- 3, - 1 - \sqrt{11}) (- 3, - 1 + \sqrt{11}), (- 3, - 1 - \sqrt{11})$

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29.

Focus $(0, 0) (0, 0)$

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31.

Foci $(- 10, 30), (- 10, - 30) (- 10, 30), (- 10, - 30)$

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33.

Center $(0, 0), (0, 0),$ Vertices $(4, 0), (- 4, 0), (0, 3), (0, - 3), (4, 0), (- 4, 0), (0, 3), (0, - 3),$ Foci $(\sqrt{7}, 0), (- \sqrt{7}, 0) (\sqrt{7}, 0), (- \sqrt{7}, 0)$

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35.

Center $(0, 0), (0, 0),$ Vertices $(\frac{1}{9}, 0), (- \frac{1}{9}, 0), (0, \frac{1}{7}), (0, - \frac{1}{7}), (\frac{1}{9}, 0), (- \frac{1}{9}, 0), (0, \frac{1}{7}), (0, - \frac{1}{7}),$ Foci $(0, \frac{4 \sqrt{2}}{63}), (0, - \frac{4 \sqrt{2}}{63}) (0, \frac{4 \sqrt{2}}{63}), (0, - \frac{4 \sqrt{2}}{63})$

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37.

Center $(- 3, 3), (- 3, 3),$ Vertices $(0, 3), (- 6, 3), (- 3, 0), (- 3, 6), (0, 3), (- 6, 3), (- 3, 0), (- 3, 6),$ Focus $(- 3, 3) (- 3, 3)$

Note that this ellipse is a circle. The circle has only one focus, which coincides with the center.

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39.

Center $(1, 1), (1, 1),$ Vertices $(5, 1), (- 3, 1), (1, 3), (1, - 1), (5, 1), (- 3, 1), (1, 3), (1, - 1),$ Foci $(1, 1 + 2 \sqrt{3}), (1, 1 - 2 \sqrt{3}) (1, 1 + 2 \sqrt{3}), (1, 1 - 2 \sqrt{3})$

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41.

Center $(- 4, 5), (- 4, 5),$ Vertices $(- 2, 5), (- 6, 4), (- 4, 6), (- 4, 4), (- 2, 5), (- 6, 4), (- 4, 6), (- 4, 4),$ Foci $(- 4 + \sqrt{3}, 5), (- 4 - \sqrt{3}, 5) (- 4 + \sqrt{3}, 5), (- 4 - \sqrt{3}, 5)$

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43.

Center $(- 2, 1), (- 2, 1),$ Vertices $(0, 1), (- 4, 1), (- 2, 5), (- 2, - 3), (0, 1), (- 4, 1), (- 2, 5), (- 2, - 3),$ Foci $(- 2, 1 + 2 \sqrt{3}), (- 2, 1 - 2 \sqrt{3}) (- 2, 1 + 2 \sqrt{3}), (- 2, 1 - 2 \sqrt{3})$

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45.

Center $(- 2, - 2), (- 2, - 2),$ Vertices $(0, - 2), (- 4, - 2), (- 2, 0), (- 2, - 4), (0, - 2), (- 4, - 2), (- 2, 0), (- 2, - 4),$ Focus $(- 2, - 2) (- 2, - 2)$

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47.

$\frac{x^{2}}{25} + \frac{y^{2}}{29} = 1 \frac{x^{2}}{25} + \frac{y^{2}}{29} = 1$

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49.

$\frac{{(x - 4)}^{2}}{25} + \frac{{(y - 2)}^{2}}{1} = 1 \frac{{(x - 4)}^{2}}{25} + \frac{{(y - 2)}^{2}}{1} = 1$

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51.

$\frac{{(x + 3)}^{2}}{16} + \frac{{(y - 4)}^{2}}{4} = 1 \frac{{(x + 3)}^{2}}{16} + \frac{{(y - 4)}^{2}}{4} = 1$

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53.

$\frac{x^{2}}{81} + \frac{y^{2}}{9} = 1 \frac{x^{2}}{81} + \frac{y^{2}}{9} = 1$

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55.

$\frac{{(x + 2)}^{2}}{4} + \frac{{(y - 2)}^{2}}{9} = 1 \frac{{(x + 2)}^{2}}{4} + \frac{{(y - 2)}^{2}}{9} = 1$

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57.

$Area = 12π square units Area = 12π$

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59.

$Area = 2 \sqrt{5} π Area = 2 \sqrt{5} π$ square units.

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61.

$Area = 9π Area = 9π$ square units.

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63.

$\frac{x^{2}}{4 h^{2}} + \frac{y^{2}}{\frac{1}{4} h^{2}} = 1 \frac{x^{2}}{4 h^{2}} + \frac{y^{2}}{\frac{1}{4} h^{2}} = 1$

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65.

$\frac{x^{2}}{400} + \frac{y^{2}}{144} = 1 \frac{x^{2}}{400} + \frac{y^{2}}{144} = 1$ . Distance = 17.32 feet

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67.

Approximately 51.96 feet

12.2 Section Exercises

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A hyperbola is the set of points in a plane the difference of whose distances from two fixed points (foci) is a positive constant.

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The foci must lie on the transverse axis and be in the interior of the hyperbola.

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The center must be the midpoint of the line segment joining the foci.

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yes $\frac{x^{2}}{6^{2}} - \frac{y^{2}}{3^{2}} = 1 \frac{x^{2}}{6^{2}} - \frac{y^{2}}{3^{2}} = 1$

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yes $\frac{x^{2}}{4^{2}} - \frac{y^{2}}{5^{2}} = 1 \frac{x^{2}}{4^{2}} - \frac{y^{2}}{5^{2}} = 1$

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11.

$\frac{x^{2}}{5^{2}} - \frac{y^{2}}{6^{2}} = 1; \frac{x^{2}}{5^{2}} - \frac{y^{2}}{6^{2}} = 1;$ vertices: $(5, 0), (- 5, 0); (5, 0), (- 5, 0);$ foci: $(\sqrt{61}, 0), (- \sqrt{61}, 0); (\sqrt{61}, 0), (- \sqrt{61}, 0);$ asymptotes: $y = \frac{6}{5} x, y = - \frac{6}{5} x y = \frac{6}{5} x, y = - \frac{6}{5} x$

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13.

$\frac{y^{2}}{2^{2}} - \frac{x^{2}}{9^{2}} = 1; \frac{y^{2}}{2^{2}} - \frac{x^{2}}{9^{2}} = 1;$ vertices: $(0, 2), (0, - 2); (0, 2), (0, - 2);$ foci: $(0, \sqrt{85}), (0, - \sqrt{85}); (0, \sqrt{85}), (0, - \sqrt{85});$ asymptotes: $y = \frac{2}{9} x, y = - \frac{2}{9} x y = \frac{2}{9} x, y = - \frac{2}{9} x$

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15.

$\frac{{(x - 1)}^{2}}{3^{2}} - \frac{{(y - 2)}^{2}}{4^{2}} = 1; \frac{{(x - 1)}^{2}}{3^{2}} - \frac{{(y - 2)}^{2}}{4^{2}} = 1;$ vertices: $(4, 2), (- 2, 2); (4, 2), (- 2, 2);$ foci: $(6, 2), (- 4, 2); (6, 2), (- 4, 2);$ asymptotes: $y = \frac{4}{3} (x - 1) + 2, y = - \frac{4}{3} (x - 1) + 2 y = \frac{4}{3} (x - 1) + 2, y = - \frac{4}{3} (x - 1) + 2$

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17.

$\frac{{(x - 2)}^{2}}{7^{2}} - \frac{{(y + 7)}^{2}}{7^{2}} = 1; \frac{{(x - 2)}^{2}}{7^{2}} - \frac{{(y + 7)}^{2}}{7^{2}} = 1;$ vertices: $(9, - 7), (- 5, - 7); (9, - 7), (- 5, - 7);$ foci: $(2 + 7 \sqrt{2}, - 7), (2 - 7 \sqrt{2}, - 7); (2 + 7 \sqrt{2}, - 7), (2 - 7 \sqrt{2}, - 7);$ asymptotes: $y = x - 9, y = - x - 5 y = x - 9, y = - x - 5$

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19.

$\frac{{(x + 3)}^{2}}{3^{2}} - \frac{{(y - 3)}^{2}}{3^{2}} = 1; \frac{{(x + 3)}^{2}}{3^{2}} - \frac{{(y - 3)}^{2}}{3^{2}} = 1;$ vertices: $(0, 3), (- 6, 3); (0, 3), (- 6, 3);$ foci: $(- 3 + 3 \sqrt{2}, 1), (- 3 - 3 \sqrt{2}, 1); (- 3 + 3 \sqrt{2}, 1), (- 3 - 3 \sqrt{2}, 1);$ asymptotes: $y = x + 6, y = - x y = x + 6, y = - x$

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21.

$\frac{{(y - 4)}^{2}}{2^{2}} - \frac{{(x - 3)}^{2}}{4^{2}} = 1; \frac{{(y - 4)}^{2}}{2^{2}} - \frac{{(x - 3)}^{2}}{4^{2}} = 1;$ vertices: $(3, 6), (3, 2); (3, 6), (3, 2);$ foci: $(3, 4 + 2 \sqrt{5}), (3, 4 - 2 \sqrt{5}); (3, 4 + 2 \sqrt{5}), (3, 4 - 2 \sqrt{5});$ asymptotes: $y = \frac{1}{2} (x - 3) + 4, y = - \frac{1}{2} (x - 3) + 4 y = \frac{1}{2} (x - 3) + 4, y = - \frac{1}{2} (x - 3) + 4$

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23.

$\frac{{(y + 5)}^{2}}{7^{2}} - \frac{{(x + 1)}^{2}}{70^{2}} = 1; \frac{{(y + 5)}^{2}}{7^{2}} - \frac{{(x + 1)}^{2}}{70^{2}} = 1;$ vertices: $(- 1, 2), (- 1, - 12); (- 1, 2), (- 1, - 12);$ foci: $(- 1, - 5 + 7 \sqrt{101}), (- 1, - 5 - 7 \sqrt{101}); (- 1, - 5 + 7 \sqrt{101}), (- 1, - 5 - 7 \sqrt{101});$ asymptotes: $y = \frac{1}{10} (x + 1) - 5, y = - \frac{1}{10} (x + 1) - 5 y = \frac{1}{10} (x + 1) - 5, y = - \frac{1}{10} (x + 1) - 5$

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25.

$\frac{{(x + 3)}^{2}}{5^{2}} - \frac{{(y - 4)}^{2}}{2^{2}} = 1; \frac{{(x + 3)}^{2}}{5^{2}} - \frac{{(y - 4)}^{2}}{2^{2}} = 1;$ vertices: $(2, 4), (- 8, 4); (2, 4), (- 8, 4);$ foci: $(- 3 + \sqrt{29}, 4), (- 3 - \sqrt{29}, 4); (- 3 + \sqrt{29}, 4), (- 3 - \sqrt{29}, 4);$ asymptotes: $y = \frac{2}{5} (x + 3) + 4, y = - \frac{2}{5} (x + 3) + 4 y = \frac{2}{5} (x + 3) + 4, y = - \frac{2}{5} (x + 3) + 4$

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27.

$y = \frac{2}{5} (x - 3) - 4, y = - \frac{2}{5} (x - 3) - 4 y = \frac{2}{5} (x - 3) - 4, y = - \frac{2}{5} (x - 3) - 4$

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29.

$y = \frac{3}{4} (x - 1) + 1, y = - \frac{3}{4} (x - 1) + 1 y = \frac{3}{4} (x - 1) + 1, y = - \frac{3}{4} (x - 1) + 1$

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31.

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33.

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35.

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37.

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39.

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41.

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43.

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45.

$\frac{x^{2}}{9} - \frac{y^{2}}{16} = 1 \frac{x^{2}}{9} - \frac{y^{2}}{16} = 1$

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47.

$\frac{{(x - 6)}^{2}}{25} - \frac{{(y - 1)}^{2}}{11} = 1 \frac{{(x - 6)}^{2}}{25} - \frac{{(y - 1)}^{2}}{11} = 1$

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49.

$\frac{{(x - 4)}^{2}}{25} - \frac{{(y - 2)}^{2}}{1} = 1 \frac{{(x - 4)}^{2}}{25} - \frac{{(y - 2)}^{2}}{1} = 1$

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51.

$\frac{y^{2}}{16} - \frac{x^{2}}{25} = 1 \frac{y^{2}}{16} - \frac{x^{2}}{25} = 1$

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53.

$\frac{y^{2}}{9} - \frac{{(x + 1)}^{2}}{9} = 1 \frac{y^{2}}{9} - \frac{{(x + 1)}^{2}}{9} = 1$

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55.

$\frac{{(x + 3)}^{2}}{25} - \frac{{(y + 3)}^{2}}{25} = 1 \frac{{(x + 3)}^{2}}{25} - \frac{{(y + 3)}^{2}}{25} = 1$

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57.

$y (x) = 3 \sqrt{x^{2} + 1}, y (x) = - 3 \sqrt{x^{2} + 1} y (x) = 3 \sqrt{x^{2} + 1}, y (x) = - 3 \sqrt{x^{2} + 1}$

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59.

$y (x) = 1 + 2 \sqrt{x^{2} + 4 x + 5}, y (x) = 1 - 2 \sqrt{x^{2} + 4 x + 5} y (x) = 1 + 2 \sqrt{x^{2} + 4 x + 5}, y (x) = 1 - 2 \sqrt{x^{2} + 4 x + 5}$

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61.

$\frac{x^{2}}{25} - \frac{y^{2}}{25} = 1 \frac{x^{2}}{25} - \frac{y^{2}}{25} = 1$

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63.

$\frac{x^{2}}{100} - \frac{y^{2}}{25} = 1 \frac{x^{2}}{100} - \frac{y^{2}}{25} = 1$

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65.

$\frac{x^{2}}{400} - \frac{y^{2}}{225} = 1 \frac{x^{2}}{400} - \frac{y^{2}}{225} = 1$

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67.

$4 (x - 1)^{2} - y 2^{2} = 16 4 (x - 1)^{2} - y 2^{2} = 16$

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69.

$\frac{{(x - h)}^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = (x - 3)^{2} - 9 y^{2} = 4 \frac{{(x - h)}^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = (x - 3)^{2} - 9 y^{2} = 4$

12.3 Section Exercises

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A parabola is the set of points in the plane that lie equidistant from a fixed point, the focus, and a fixed line, the directrix.

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The graph will open down.

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The distance between the focus and directrix will increase.

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yes $x^{2} = 4 (\frac{1}{16}) y x^{2} = 4 (\frac{1}{16}) y$

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yes ${(y - 3)}^{2} = 4 (2) (x - 2) {(y - 3)}^{2} = 4 (2) (x - 2)$

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11.

$y^{2} = \frac{1}{8} x, V : (0, 0); F : (\frac{1}{32}, 0); d : x = - \frac{1}{32} y^{2} = \frac{1}{8} x, V : (0, 0); F : (\frac{1}{32}, 0); d : x = - \frac{1}{32}$

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13.

$x^{2} = - \frac{1}{4} y, V : (0, 0); F : (0, - \frac{1}{16}); d : y = \frac{1}{16} x^{2} = - \frac{1}{4} y, V : (0, 0); F : (0, - \frac{1}{16}); d : y = \frac{1}{16}$

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15.

$y^{2} = \frac{1}{36} x, V : (0, 0); F : (\frac{1}{144}, 0); d : x = - \frac{1}{144} y^{2} = \frac{1}{36} x, V : (0, 0); F : (\frac{1}{144}, 0); d : x = - \frac{1}{144}$

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17.

${(x - 1)}^{2} = 4 (y - 1), V : (1, 1); F : (1, 2); d : y = 0 {(x - 1)}^{2} = 4 (y - 1), V : (1, 1); F : (1, 2); d : y = 0$

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19.

${(y - 4)}^{2} = 2 (x + 3), V : (- 3, 4); F : (- \frac{5}{2}, 4); d : x = - \frac{7}{2} {(y - 4)}^{2} = 2 (x + 3), V : (- 3, 4); F : (- \frac{5}{2}, 4); d : x = - \frac{7}{2}$

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21.

${(x + 4)}^{2} = 24 (y + 1), V : (- 4, - 1); F : (- 4, 5); d : y = −7 {(x + 4)}^{2} = 24 (y + 1), V : (- 4, - 1); F : (- 4, 5); d : y = −7$

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23.

${(y - 3)}^{2} = −12 (x + 1), V : (- 1, 3); F : (- 4, 3); d : x = 2 {(y - 3)}^{2} = −12 (x + 1), V : (- 1, 3); F : (- 4, 3); d : x = 2$

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25.

${(x - 5)}^{2} = \frac{4}{5} (y + 3), V : (5, - 3); F : (5, - \frac{14}{5}); d : y = - \frac{16}{5} {(x - 5)}^{2} = \frac{4}{5} (y + 3), V : (5, - 3); F : (5, - \frac{14}{5}); d : y = - \frac{16}{5}$

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27.

${(x - 2)}^{2} = −2 (y - 5), V : (2, 5); F : (2, \frac{9}{2}); d : y = \frac{11}{2} {(x - 2)}^{2} = −2 (y - 5), V : (2, 5); F : (2, \frac{9}{2}); d : y = \frac{11}{2}$

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29.

${(y - 1)}^{2} = \frac{4}{3} (x - 5), V : (5, 1); F : (\frac{16}{3}, 1); d : x = \frac{14}{3} {(y - 1)}^{2} = \frac{4}{3} (x - 5), V : (5, 1); F : (\frac{16}{3}, 1); d : x = \frac{14}{3}$

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31.

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33.

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35.

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37.

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39.

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41.

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43.

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45.

$x^{2} = −16 y x^{2} = −16 y$

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47.

${(y - 2)}^{2} = 4 \sqrt{2} (x - 2) {(y - 2)}^{2} = 4 \sqrt{2} (x - 2)$

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49.

${(y + \sqrt{3})}^{2} = −4 \sqrt{2} (x - \sqrt{2}) {(y + \sqrt{3})}^{2} = −4 \sqrt{2} (x - \sqrt{2})$

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51.

$x^{2} = y x^{2} = y$

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53.

${(y - 2)}^{2} = \frac{1}{4} (x + 2) {(y - 2)}^{2} = \frac{1}{4} (x + 2)$

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55.

${(y - \sqrt{3})}^{2} = 4 \sqrt{5} (x + \sqrt{2}) {(y - \sqrt{3})}^{2} = 4 \sqrt{5} (x + \sqrt{2})$

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57.

$y^{2} = −8 x y^{2} = −8 x$

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59.

${(y + 1)}^{2} = 12 (x + 3) {(y + 1)}^{2} = 12 (x + 3)$

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61.

$(0, 1) (0, 1)$

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63.

At the point 2.25 feet above the vertex.

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65.

0.5625 feet

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67.

$x^{2} = −125 (y - 20), x^{2} = −125 (y - 20),$ height is 7.2 feet

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69.

2304 feet

12.4 Section Exercises

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The $x y x y$ term causes a rotation of the graph to occur.

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The conic section is a hyperbola.

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It gives the angle of rotation of the axes in order to eliminate the $x y x y$ term.

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$A B = 0, A B = 0,$ parabola

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$A B = - 4 < 0, A B = - 4 < 0,$ hyperbola

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11.

$A B = 6 > 0, A B = 6 > 0,$ ellipse

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13.

$B^{2} - 4 A C = 0, B^{2} - 4 A C = 0,$ parabola

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15.

$B^{2} - 4 A C = 0, B^{2} - 4 A C = 0,$ parabola

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17.

$B^{2} - 4 A C = - 96 < 0, B^{2} - 4 A C = - 96 < 0,$ ellipse

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19.

$7 {x^{'}}^{2} + 9 {y^{'}}^{2} - 4 = 0 7 {x^{'}}^{2} + 9 {y^{'}}^{2} - 4 = 0$

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21.

$3 {x^{'}}^{2} + 2 x^{'} y^{'} - 5 {y^{'}}^{2} + 1 = 0 3 {x^{'}}^{2} + 2 x^{'} y^{'} - 5 {y^{'}}^{2} + 1 = 0$

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23.

$θ = 60^{\circ}, 11 {x^{'}}^{2} - {y^{'}}^{2} + \sqrt{3} x^{'} + y^{'} - 4 = 0 θ = 60^{\circ}, 11 {x^{'}}^{2} - {y^{'}}^{2} + \sqrt{3} x^{'} + y^{'} - 4 = 0$

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25.

$θ = 150^{\circ}, 21 {x^{'}}^{2} + 9 {y^{'}}^{2} + 4 x^{'} - 4 \sqrt{3} y^{'} - 6 = 0 θ = 150^{\circ}, 21 {x^{'}}^{2} + 9 {y^{'}}^{2} + 4 x^{'} - 4 \sqrt{3} y^{'} - 6 = 0$

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27.

$θ \approx {36.9}^{\circ}, 125 {x^{'}}^{2} + 6 x^{'} - 42 y^{'} + 10 = 0 θ \approx {36.9}^{\circ}, 125 {x^{'}}^{2} + 6 x^{'} - 42 y^{'} + 10 = 0$

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29.

$θ = 45^{\circ}, 3 {x^{'}}^{2} - {y^{'}}^{2} - \sqrt{2} x^{'} + \sqrt{2} y^{'} + 1 = 0 θ = 45^{\circ}, 3 {x^{'}}^{2} - {y^{'}}^{2} - \sqrt{2} x^{'} + \sqrt{2} y^{'} + 1 = 0$

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31.

$\frac{\sqrt{2}}{2} (x^{'} + y^{'}) = \frac{1}{2} {(x^{'} - y^{'})}^{2} \frac{\sqrt{2}}{2} (x^{'} + y^{'}) = \frac{1}{2} {(x^{'} - y^{'})}^{2}$

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33.

$\frac{{(x^{'} - y^{'})}^{2}}{8} + \frac{{(x^{'} + y^{'})}^{2}}{2} = 1 \frac{{(x^{'} - y^{'})}^{2}}{8} + \frac{{(x^{'} + y^{'})}^{2}}{2} = 1$

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35.

$\frac{{(x^{'} + y^{'})}^{2}}{2} - \frac{{(x^{'} - y^{'})}^{2}}{2} = 1 \frac{{(x^{'} + y^{'})}^{2}}{2} - \frac{{(x^{'} - y^{'})}^{2}}{2} = 1$

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37.

$\frac{\sqrt{3}}{2} x^{'} - \frac{1}{2} y^{'} = {(\frac{1}{2} x^{'} + \frac{\sqrt{3}}{2} y^{'} - 1)}^{2} \frac{\sqrt{3}}{2} x^{'} - \frac{1}{2} y^{'} = {(\frac{1}{2} x^{'} + \frac{\sqrt{3}}{2} y^{'} - 1)}^{2}$

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39.

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41.

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43.

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45.

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47.

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49.

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51.

$θ = 45^{\circ} θ = 45^{\circ}$

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53.

$θ = 60^{\circ} θ = 60^{\circ}$

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55.

$θ \approx {36.9}^{\circ} θ \approx {36.9}^{\circ}$

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57.

$- 4 \sqrt{6} < k < 4 \sqrt{6} - 4 \sqrt{6} < k < 4 \sqrt{6}$

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59.

$k = 2 k = 2$

12.5 Section Exercises

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If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a hyperbola.

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The directrix will be parallel to the polar axis.

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One of the foci will be located at the origin.

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Parabola with $e = 1 e = 1$ and directrix $\frac{3}{4} \frac{3}{4}$ units below the pole.

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Hyperbola with $e = 2 e = 2$ and directrix $\frac{5}{2} \frac{5}{2}$ units above the pole.

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11.

Parabola with $e = 1 e = 1$ and directrix $\frac{3}{10} \frac{3}{10}$ units to the right of the pole.

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13.

Ellipse with $e = \frac{2}{7} e = \frac{2}{7}$ and directrix $22$ units to the right of the pole.

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15.

Hyperbola with $e = \frac{5}{3} e = \frac{5}{3}$ and directrix $\frac{11}{5} \frac{11}{5}$ units above the pole.

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17.

Hyperbola with $e = \frac{8}{7} e = \frac{8}{7}$ and directrix $\frac{7}{8} \frac{7}{8}$ units to the right of the pole.

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19.

$25 x^{2} + 16 y^{2} - 12 y - 4 = 0 25 x^{2} + 16 y^{2} - 12 y - 4 = 0$

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21.

$21 x^{2} - 4 y^{2} - 30 x + 9 = 0 21 x^{2} - 4 y^{2} - 30 x + 9 = 0$

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23.

$64 y^{2} = 48 x + 9 64 y^{2} = 48 x + 9$

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25.

$96 y^{2} - 25 x^{2} + 110 y + 25 = 0 96 y^{2} - 25 x^{2} + 110 y + 25 = 0$

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27.

$3 x^{2} + 4 y^{2} - 2 x - 1 = 0 3 x^{2} + 4 y^{2} - 2 x - 1 = 0$

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29.

$5 x^{2} + 9 y^{2} - 24 x - 36 = 0 5 x^{2} + 9 y^{2} - 24 x - 36 = 0$

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31.

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33.

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35.

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37.

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39.

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41.

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43.

$r = \frac{4}{5 + \cos θ} r = \frac{4}{5 + \cos θ}$

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45.

$r = \frac{4}{1 + 2 \sin θ} r = \frac{4}{1 + 2 \sin θ}$

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47.

$r = \frac{1}{1 + \cos θ} r = \frac{1}{1 + \cos θ}$

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49.

$r = \frac{7}{8 - 28 \cos θ} r = \frac{7}{8 - 28 \cos θ}$

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51.

$r = \frac{12}{2 + 3 \sin θ} r = \frac{12}{2 + 3 \sin θ}$

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53.

$r = \frac{15}{4 - 3 \cos θ} r = \frac{15}{4 - 3 \cos θ}$

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55.

$r = \frac{3}{3 - 3 \cos θ} r = \frac{3}{3 - 3 \cos θ}$

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57.

$r = \pm \frac{2}{\sqrt{1 + \sin θ \cos θ}} r = \pm \frac{2}{\sqrt{1 + \sin θ \cos θ}}$

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59.

$r = \pm \frac{2}{4 \cos θ + 3 \sin θ} r = \pm \frac{2}{4 \cos θ + 3 \sin θ}$

Review Exercises

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$\frac{x^{2}}{5^{2}} + \frac{y^{2}}{8^{2}} = 1; \frac{x^{2}}{5^{2}} + \frac{y^{2}}{8^{2}} = 1;$ center: $(0, 0); (0, 0);$ vertices: $(5, 0), (−5, 0), (0, 8), (0, - 8); (5, 0), (−5, 0), (0, 8), (0, - 8);$ foci: $(0, \sqrt{39}), (0, - \sqrt{39}) (0, \sqrt{39}), (0, - \sqrt{39})$

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$\frac{{(x + 3)}^{2}}{1^{2}} + \frac{{(y - 2)}^{2}}{3^{2}} = 1 (- 3, 2); (- 2, 2), (- 4, 2), (- 3, 5), (- 3, - 1); (- 3, 2 + 2 \sqrt{2}), (- 3, 2 - 2 \sqrt{2}) \frac{{(x + 3)}^{2}}{1^{2}} + \frac{{(y - 2)}^{2}}{3^{2}} = 1$

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center: $(0, 0); (0, 0);$ vertices: $(6, 0), (−6, 0), (0, 3), (0, −3); (6, 0), (−6, 0), (0, 3), (0, −3);$ foci: $(3 \sqrt{3}, 0), (- 3 \sqrt{3}, 0) (3 \sqrt{3}, 0), (- 3 \sqrt{3}, 0)$

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center: $(−2, −2); (−2, −2);$ vertices: $(2, −2), (−6, −2), (−2, 6), (−2, −10); (2, −2), (−6, −2), (−2, 6), (−2, −10);$ foci: $(−2, −2 + 4 \sqrt{3},), (−2, −2 −4 \sqrt{3}) (−2, −2 + 4 \sqrt{3},), (−2, −2 −4 \sqrt{3})$

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$\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1 \frac{x^{2}}{25} + \frac{y^{2}}{16} = 1$

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11.

Approximately 35.71 feet

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13.

$\frac{{(y + 1)}^{2}}{4^{2}} - \frac{{(x - 4)}^{2}}{6^{2}} = 1; \frac{{(y + 1)}^{2}}{4^{2}} - \frac{{(x - 4)}^{2}}{6^{2}} = 1;$ center: $(4, −1); (4, −1);$ vertices: $(4, 3), (4, −5); (4, 3), (4, −5);$ foci: $(4, −1 + 2 \sqrt{13}), (4, −1 - 2 \sqrt{13}) (4, −1 + 2 \sqrt{13}), (4, −1 - 2 \sqrt{13})$

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15.

$\frac{{(x - 2)}^{2}}{2^{2}} - \frac{{(y + 3)}^{2}}{{(2 \sqrt{3})}^{2}} = 1; \frac{{(x - 2)}^{2}}{2^{2}} - \frac{{(y + 3)}^{2}}{{(2 \sqrt{3})}^{2}} = 1;$ center: $(2, −3); (2, −3);$ vertices: $(4, −3), (0, −3); (4, −3), (0, −3);$ foci: $(6, −3), (−2, −3) (6, −3), (−2, −3)$

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17.

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19.

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21.

$\frac{{(x - 5)}^{2}}{1} - \frac{{(y - 7)}^{2}}{3} = 1 \frac{{(x - 5)}^{2}}{1} - \frac{{(y - 7)}^{2}}{3} = 1$

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23.

${(x + 2)}^{2} = \frac{1}{2} (y - 1); {(x + 2)}^{2} = \frac{1}{2} (y - 1);$ vertex: $(−2, 1); (−2, 1);$ focus: $(−2, \frac{9}{8}); (−2, \frac{9}{8});$ directrix: $y = \frac{7}{8} y = \frac{7}{8}$

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25.

${(x + 5)}^{2} = (y + 2); {(x + 5)}^{2} = (y + 2);$ vertex: $(- 5, - 2); (- 5, - 2);$ focus: $(- 5, - \frac{7}{4}); (- 5, - \frac{7}{4});$ directrix: $y = - \frac{9}{4} y = - \frac{9}{4}$

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27.

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29.

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31.

${(x - 2)}^{2} = (\frac{1}{2}) (y - 1) {(x - 2)}^{2} = (\frac{1}{2}) (y - 1)$

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33.

$B^{2} - 4 A C = 0, B^{2} - 4 A C = 0,$ parabola

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35.

$B^{2} - 4 A C = - 31 < 0, B^{2} - 4 A C = - 31 < 0,$ ellipse

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37.

$θ = 45^{\circ}, {x^{'}}^{2} + 3 {y^{'}}^{2} - 12 = 0 θ = 45^{\circ}, {x^{'}}^{2} + 3 {y^{'}}^{2} - 12 = 0$

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39.

$θ = 45^{\circ} θ = 45^{\circ}$

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41.

Hyperbola with $e = 5 e = 5$ and directrix $22$ units to the left of the pole.

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43.

Ellipse with $e = \frac{3}{4} e = \frac{3}{4}$ and directrix $\frac{1}{3} \frac{1}{3}$ unit above the pole.

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45.

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47.

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49.

$r = \frac{3}{1 + \cos θ} r = \frac{3}{1 + \cos}$

Practice Test

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$\frac{x^{2}}{3^{2}} + \frac{y^{2}}{2^{2}} = 1; \frac{x^{2}}{3^{2}} + \frac{y^{2}}{2^{2}} = 1;$ center: $(0, 0); (0, 0);$ vertices: $(3, 0), (–3, 0), (0, 2), (0, −2); (3, 0), (–3, 0), (0, 2), (0, −2);$ foci: $(\sqrt{5}, 0), (- \sqrt{5}, 0) (\sqrt{5}, 0), (- \sqrt{5}, 0)$

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center: $(3, 2); (3, 2);$ vertices: $(11, 2), (−5, 2), (3, 8), (3, −4); (11, 2), (−5, 2), (3, 8), (3, −4);$ foci: $(3 + 2 \sqrt{7}, 2), (3 - 2 \sqrt{7}, 2) (3 + 2 \sqrt{7}, 2), (3 - 2 \sqrt{7}, 2)$

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$\frac{{(x - 1)}^{2}}{36} + \frac{{(y - 2)}^{2}}{27} = 1 \frac{{(x - 1)}^{2}}{36} + \frac{{(y - 2)}^{2}}{27} = 1$

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$\frac{x^{2}}{7^{2}} - \frac{y^{2}}{9^{2}} = 1; \frac{x^{2}}{7^{2}} - \frac{y^{2}}{9^{2}} = 1;$ center: $(0, 0); (0, 0);$ vertices $(7, 0), (−7, 0); (7, 0), (−7, 0);$ foci: $(\sqrt{130}, 0), (- \sqrt{130}, 0); (\sqrt{130}, 0), (- \sqrt{130}, 0);$ asymptotes: $y = \pm \frac{9}{7} x y = \pm \frac{9}{7} x$

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center: $(3, −3); (3, −3);$ vertices: $(8, −3), (−2, −3); (8, −3), (−2, −3);$ foci: $(3 + \sqrt{26}, −3), (3 - \sqrt{26}, −3); (3 + \sqrt{26}, −3), (3 - \sqrt{26}, −3);$ asymptotes: $y = \pm \frac{1}{5} (x - 3) - 3 y = \pm \frac{1}{5} (x - 3) - 3$

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11.

$\frac{{(y - 3)}^{2}}{1} - \frac{{(x - 1)}^{2}}{8} = 1 \frac{{(y - 3)}^{2}}{1} - \frac{{(x - 1)}^{2}}{8} = 1$

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13.

${(x - 2)}^{2} = \frac{1}{3} (y + 1); {(x - 2)}^{2} = \frac{1}{3} (y + 1);$ vertex: $(2, −1); (2, −1);$ focus: $(2, - \frac{11}{12}); (2, - \frac{11}{12});$ directrix: $y = - \frac{13}{12} y = - \frac{13}{12}$

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15.

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17.

Approximately $8.49 8.49$ feet

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19.

parabola; $θ \approx {63.4}^{\circ} θ \approx {63.4}^{\circ}$

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21.

${x^{'}}^{2} - 4 x^{'} + 3 y^{'} = 0 {x^{'}}^{2} - 4 x^{'} + 3 y^{'} = 0$

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23.

Hyperbola with $e = \frac{3}{2}, e = \frac{3}{2},$ and directrix $\frac{5}{6} \frac{5}{6}$ units to the right of the pole.

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25.