38. Think About a Plan The light emitted from a lamp with a shade forms a shadow on the wall. How can you turn the lamp in relation to the wall so that the shadow cast by the shade forms a parabola and a circle?

    • How can a drawing or model help you solve this problem?

    • Can you form a hyperbola and an ellipse? If so, explain how.

      

39. 1. Writing Describe the relationship between the center of a circle and the axes of symmetry of the circle.
    2. Make a Conjecture Where is the center of an ellipse or a hyperbola located in relation to the axes of symmetry? Verify your conjecture with examples.

Graph each circle with the given radius or diameter so that the center is at the origin. Then write the equation for each graph.

40. radius 6
41. radius 1 2 1 half
42. diameter 8
43. diameter 2.5

Mental Math Each given point is on the graph of the given equation. Use symmetry to find at least one more point on the graph.

44. ( 2 , − 4 ) , y 2 = 8 x open 2 comma negative 4 close comma . y squared , equals 8 . x
45. ( − 2 , 1 ) , x 2 + y 2 = 3 open . negative square root of 2 comma 1 . close . comma , x squared , plus , y squared , equals 3
46. ( 2 , 2 2 ) , x 2 + 4 y 2 = 36 open . 2 comma 2 square root of 2 . close . comma , x squared , plus 4 , y squared , equals 36
47. ( − 2 , 0 ) , 9 x 2 + 9 y 2 − 36 = 0 negative 2 comma 0 close comma , 9 x squared , plus , 9 y squared , minus 36 equals 0
48. ( − 3 , − 51 ) , 6 y 2 − 9 x 2 − 225 = 0 open . negative 3 comma negative square root of 51 . close . comma 6 , y squared , minus 9 , x squared , minus 225 equals 0
49. ( 0 , 7 ) , x 2 + 2 y 2 = 14 open , 0 comma square root of 7 , close . comma , x squared , plus 2 , y squared , equals 14

50. Sound An airplane flying faster than the speed of sound creates a cone-shaped pressure disturbance in the air. This is heard by people on the ground as a sonic boom. What is the shape of the path on the ground?

    

51. Open-Ended Describe any other figures you can see that can be formed by the intersection of a plane and another shape, such as a sphere.

C Challenge

52. 1. Graph the equation x y = 16 . x y equals 16 .  Use both positive and negative values for x.
    2. Which conic section does the equation appear to model?
    3. Identify any intercepts and lines of symmetry.
    4. Does your graph represent a function? If so, rewrite the equation using function notation.
53. Graphing Calculator An xy-term has an interesting effect on the graph of a conic section. Sketch the graph of each conic section below using your graphing calculator. (Hint: To solve for y, you will need to complete a square.)
    1. 4 x 2 + 2 x y + y 2 = 9 4 x squared , plus 2 x y plus . y squared , equals 9
    2. 4 x 2 + 2 x y − y 2 = 9 4 x squared , plus 2 x y minus . y squared , equals 9

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