HYPERBOLA. Going off on a TANGENT!

Transcription

1 HYPERBOLA Going off on a TANGENT!

2 RECALL THAT THE HYPERBOLA IS A CONIC SECTION

3 A LAMP CASTS A HYPERBOLIC BEAM OF LIGHT

4 NUCLEAR COOLING TOWERS

5 TORNADO TOWER, QATAR

6 KOBE PORT TOWER, JAPAN

7 RULED HYPERBOLOID

8 MEET THE HYPERBOLA

9 HOW IS A HYPERBOLA LIKE AN ELLIPSE? Hmmm..

10 NOT REALLY AT ALL...AT LEAST NOT TO LOOK AT THEM. Foci and vertices on the transverse axis Curve does not touch the conjugate axis There are no vertices on the conjugate axis Pythagorean relationship to find foci

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13 Play around with an interactive hyperbola or build your own with thumbtacks and string.

14 Think about the interactive demonstration about how a hyperbola was created. In the interactive hyperbola the difference between the length of the green segment and the red segment remained constant. You should have an idea of what that constant may be?

15 The difference between the lengths of the two segments is equal to the length of the transverse axis Transverse axis has the foci and vertices. Conjugate axis is the other one A hyperbola does NOT always have one axis bigger than another they can be the same length these are called equilateral hyperbolas

16 USE P.T. TO DETERMINE THE COORDINATES OF THE FOCI The distance of the focus to the center is c a 2 + b 2 = c 2

17 THE TRANSVERSE AXIS CAN ALSO BE ON THE VERTICAL AXIS---this is allowed J

18 SO NOW WHAT DOES THE EQUATION OF A HYPERBOLA LOOK LIKE? Recall that an ellipse considers that it is the SUM of the two segments that remains constant..a HYPERBOLA considers that it is the DIFFERENCE between the two segments that remains constant Example x 2 25 y 2 9 =1

19 THERE ARE ACTUALLY TWO WAYS TO WRITE THE EQUATION OF A HYPERBOLA x 2 a 2 y 2 b 2 =1 x 2 a 2 y 2 b 2 = 1 If this is POSITIVE 1 If this is NEGATIVE 1 Horizontal Transverse axis Vertical Transverse axis

20 Case #1 TRANSVERSE AXIS is HORIZONTAL x 2 25 y 2 9 =1 We know that right away because the equation is equal to POSITIVE 1 What is under the x 2 determines intercepts along the transverse axis we need to square root 25 to get the x intercepts What is under the y 2 determines the length of the conjugate axis and helps us to draw the ASYMPTOTES. It is NOT a point on the curve

21 Now what the HECK does that mean??

22 IT MEANS YOU NEED TO DRAW THE ASYMPTOTES BEFORE YOU DRAW THE CURVE Draw a box using the values of a and b from the equation Draw lines (asymptotes) through the corners to guide your curve Locate the vertices on the transverse axis and a few other points to draw curve

23 AND OF COURSE, ASYMPTOTES HAVE THEIR OWN EQUATIONS: y = ± b a x ü There are 2 ü They are straight ü They pass through origin ü One has positive slope ü One has negative slope

24 LIKE THIS

25 Case #2 CONJUGATE AXIS is VERTICAL x 2 49 y 2 16 = 1 We know that right away because the equation is equal to NEGATIVE 1 What is under the x 2 determines the length of the conjugate axis and helps us to draw the ASYMPTOTES we need to square root 49. What is under the y 2 determines intercepts along the transverse axis we need to square root 16 to get the y-intercepts

26 JUST A QUICKIE x y 2 64 = 1 solution a 2 =100 a = 100 a =10 b 2 = 64 b = 64 b = 8 The length of the transverse axis is 20 and the length of the conjugate axis is 16 and the foci are on the vertical axis

27 SO DRAW THE ASYMPTOTES

28 PUTTING IT ALL TOGETHER x 2 49 y solution So the foci are =1 What are the foci and sketch it? a 2 = 49 a = 49 a = 7 ( 170,0) and b 2 =121 b = 121 b =11 ( 170,0) a 2 + b 2 = c = c 2 c = c = 170

29 AND IT LOOKS LIKE THIS:

30 APPLICATION LOCATING POSITION If the sound of an enemy gun is heard at two listening posts and the difference in time is calculated, then the gun is known to be located on a particular hyperbola. A third listening post will determine a second hyperbola, and then the gun placement can be found at the intersection of the two hyperbolas.

31 LET S LOOK AT PAGE 340 #5 TOGETHER HOMEWORK WB page 339 #2, #4 b) c) #6 a) d) e) f) #7 d) #9 d) #15, #18, #20 Please be sure to check your answers on the class website before next class. That way you can come prepared with your concerns and questions.