# Hyperbolas

Identify the type of conic section (parabola, ellipse,

circle, or hyperbola) represented by each of the

following equations. (In the case of a circle, identify

the conic section as a circle rather than an ellipse.) Do

NOT write the equations in standard form; these

questions can instead be answered by looking at the

signs of the quadratic terms .

1. 2y + x^{2} + 9x = 0

2. 14x^{2} + 7x −12y = −6y^{2} + 95

3. 7x^{2} − 3y^{2} = 5x − y + 40

4. y^{2} + 9 = 9y − x

5. 3x^{2} − 7x + 3y^{2} = −12y +13

6. x^{2} +10x = −2y − y^{2} + 5

7. 4y^{2} + 2x^{2} = 8y − 6x + 9

8. 8y^{2} + 24x = 8x^{2} + 30

Write each of the following equations in the standard

form for the equation of a hyperbola, where the

standard form is represented by one of the following

equations:

9. y^{2} −8x^{2} −8 = 0

10. 3x^{2} −10y^{2} − 30 = 0

11. x^{2} − y^{2} − 6x = −2y − 3

12. 9x^{2} − 3y^{2} = 48y +192

13. 7x^{2} − 5y^{2} +14x + 20y − 48 = 0

14. 9y^{2} − 2x^{2} + 90y +16x +175 = 0

**Answer the following.**

15. The length of the transverse axis of a hyperbola

is __________.

16. The length of the conjugate axis of a hyperbola is

__________.

17. The following questions establish the formulas

for the slant asymptotes of

(a) State the point- slope equation for a line.

(b) Substitute the center of the hyperbola,

(h, k ) into the equation from part (a).

(c) Recall that the formula for slope is

represented by . In the equation

, what is the “rise” of

each slant asymptote from the center? What

is the “run” of each slant asymptote from the

center?

(d) Based on the answers to part (c), what is the

slope of each of the asymptotes for the graph

of ? (Remember that

there are two slant asymptotes passing

through the center of the hyperbola, one

having positive slope and one having

negative slope .)

(e) Substitute the slopes from part (d) into the

equation from part (b) to obtain the

equations of the slant asymptotes.

18. The following questions establish the formulas

for the slant asymptotes of

(a) State the point-slope equation for a line .

(b) Substitute the center of the hyperbola,

(h, k ) into the equation from part (a).

(c) Recall that the formula for slope is

represented by . In the equation

, what is the “rise” of

each slant asymptote from the center? What

is the “run” of each slant asymptote from the

center?

(d) Based on the answers to part (c), what is the

slope of each of the asymptotes for the graph

of ? (Remember that

there are two slant asymptotes passing

through the center of the hyperbola, one

having positive slope and one having

negative slope.)

(e) Substitute the slopes from part (d) into the

equation from part (b) to obtain the

equations of the slant asymptotes.

19. In the standard form for the equation of a

hyperbola, a^{2} represents (choose one):

the larger denominator

the denominator of the first term

20. In the standard form for the equation of a

hyperbola, b^{2} represents (choose one):

the smaller denominator

the denominator of the second term

**Answer the following for each hyperbola. For answers
involving radicals , give exact answers and then round
to the nearest tenth.**

**(a) Write the given equation in the standard form
for the equation of a hyperbola. (Some
equations may already be given in standard
form.)**

It may be helpful to begin sketching the graph for

part (h) as a visual aid to answer the questions

below.

(b) State the coordinates of the center .

(c) State the coordinates of the vertices, and then

state the length of the transverse axis.

(d) State the coordinates of the endpoints of the

conjugate axis, and then state the length of the

conjugate axis.

(e) State the coordinates of the foci.

(f) State the equations of the asymptotes.

(Answers may be left in point-slope form.)

(g) State the eccentricity.

(h) Sketch a graph of the hyperbola which

includes the features from (b)-(f), along with

the central rectangle. Label the center C, the

vertices V_{1} and V_{2}, and the foci F_{1} and F_{2}.

29. x^{2} − 25y^{2} + 8x −150y − 234 = 0

30. 4y^{2} −81x^{2} = −162x + 405

31. 64x^{2} − 9y^{2} +18y = 521−128x

32. 16x^{2} − 9y^{2} − 64x −18y −89 = 0

33. 5y^{2} − 4x^{2} − 50y − 24x + 69 = 0

34. 7x^{2} − 9y^{2} − 72y = 32 − 70x

35. x^{2} − 3y^{2} = 18x + 27

36. 4y^{2} − 21x^{2} −8y − 42x − 89 = 0

Use the given features of each of the following

hyperbolas to write an equation for the hyperbola in

standard form.

37. Center: (0, 0)

a = 8

b = 5

Horizontal Transverse Axis

38. Center: (0, 0)

a = 7

b = 3

Vertical Transverse Axis

39. Center: (−2, − 5)

a = 2

b =10

Vertical Transverse Axis

40. Center: (3, − 4)

a =1

b = 6

Horizontal Transverse Axis

41. Center: (−6,1)

Length of transverse axis: 10

Length of conjugate axis: 8

Vertical Transverse Axis

42. Center: (2, 5)

Length of transverse axis: 6

Length of conjugate axis: 14

Horizontal Transverse Axis

43. Foci: (0, 9) and (0, − 9)

Length of transverse axis: 6

44. Foci: (5, 0) and (−5, 0)

Length of conjugate axis: 4

45. Foci: (−2, 3) and (10, 3)

Length of conjugate axis: 10

46. Foci: (−3, 8) and (−3, − 6)

Length of transverse axis: 8

47. Vertices: (4, − 7) and (4, 9)

b = 4

48. Vertices: (1, 6) and (7, 6)

b = 7

49. Center: (5, 3)

One focus is at (5, 8)

One vertex is at (5, 6)

50. Center: (−3, − 4)

One focus is at (7, − 4)

One vertex is at (5, − 4)

51. Center: (−1, 2)

Vertex: (3, 2)

Equation of one asymptote:

7x − 4y = −15

52. Center: (−1, 2)

Vertex: (3, 2)

Equation of one asymptote:

6x + 5y = −7

53. Vertices: (−1, 4) and (7, 4)

e = 3

54. Vertices: (2, 6) and (2, −1)

55. Center: (4, − 3)

One focus is at (4, 6)

56. Center: (−1, − 2)

One focus is at (9, − 2)

57. Foci: (3, 0) and (3, 8)

58. Foci: (−6, − 5) and (−6, 7)

e = 2

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