Write an equation of a hyperbola from the given information. Assume the center of each hyperbola is (0, 0).

29. Transverse axis is vertical and is 9 units; central rectangle is 9 units by 4 units
30. Perimeter of central rectangle is 16 units; vertices are (0, 3) and ( 0 , − 3 ) open 0 comma negative 3 close
31. ( Distance from the center of a hyperbola to a focus ) 2 = 96 ; open . cap distancefromthecenterofahyperbolatoafocus . close squared . equals 96 semicolon  endpoints of the transverse axis are at ( − 32 , 0 ) open . negative square root of 32 comma 0 . close  and ( 32 , 0 ) . open , square root of 32 comma 0 , close . .

Graphing Calculator Solve each equation for y. Graph each relation on your graphing calculator. Use the TRACE feature to locate the vertices.

32. x 2 − 2 y 2 = 4 x squared , minus , 2 y squared , equals 4
33. x 2 − y 2 = 1 x squared , minus . y squared , equals 1
34. 3 x 2 − y 2 = 2 3 x squared , minus . y squared , equals 2

Graph each equation.

35. 5 x 2 − 12 y 2 = 120 5 x squared , minus , 12 y squared , equals 120
36. 16 x 2 − 20 y 2 = 560 16 x squared , minus , 20 y squared , equals 560
37. y 2 20 − x 2 5 = 1 fraction y squared , over 20 end fraction , minus , fraction x squared , over 5 end fraction , equals 1
38. Comets The path of a comet around the sun followed one branch of a hyperbola. Find an equation that models its path around the sun, given that a = 40 eh equals 40  million miles and c = 250 c equals 250  million miles. Use the horizontal model.
39. Open-Ended Choose two points on an axis to be the vertices of a hyperbola. Choose two other points on the same axis to be the foci. Write the equation of your hyperbola and draw its graph.
40. Error Analysis On a test, a student found that the foci of the hyperbola with equation y 2 100 − x 2 21 = 1 fraction y squared , over 100 end fraction . minus , fraction x squared , over 21 end fraction , equals 1  were ( 0 , ± 79 ) . open . 0 comma plus minus square root of 79 . close . .  The teacher credited the student three points out of a possible five. What did the student do right? What did the student do wrong?

C Challenge

41. The function y = x 2 − 9 y equals . square root of x squared , minus 9 end root  represents part of a hyperbola. The tables show the coordinates of several points on the graph.

    
    Image Long Description

    1. Explain why ERROR appears for some entries.
    2. Describe the relationship between the x- and y-coordinates as x increases.
    3. Reasoning Do you think that the x- and y-coordinates will ever be equal? Explain.
    4. Make a Conjecture What are the equations of the asymptotes of this hyperbola? Verify your answer by drawing the complete graph.

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