Constructing Hyperbolas

1. Constructing Hyperbolas

2. A hyperbola is created from the intersection of a plane with a double cone.

3. 5 mm An example of a hyperbola occurring in nature is shown in the picture. Two parallel glass plates touching at the left end but with an opening of about 5 mm between them at the right, are dipped in beet juice. The juice rises by capillarity to form a hyperbola. The reason that the juice rises is due to the surface tension of the liquid and this surface tension counteracts the force of gravity. The closer the glass is together the more the surface tension is able to counteract gravity. As you see, as the separation between the glass increases the lower the beet juice falls.

4. A hyperbola is defined by a group of points that have a same difference of distance from two foci. When you subtract the small line from the long line for each ordered pair the remaining value is the same. Hyperbolas can be symmetrical around the x-axis or the y-axis The one on the right is symmetrical around the x-axis.

5. The hyperbola is defined by two foci. A transverse axis passes through both foci. A conjugate axis is perpendicular to the transverse axis through the centre (0,0). There are two asymptotes that intersect each other through the centre. They restrict the path of the hyperbola. The form of this horizontal hyperbola is:

6. Asymptotes: Asymptote: Vertices: (a,0) and (-a,0) c2 = a2 + b2 Foci: (c,0) and (-c,0) Vertices: (0,a) and (0,-a) c2 = a2 + b2 Foci: (0,c) and (0,-c)

7. Asymptote: Asymptote: a2 = 16 a = 4 b2 = 9 b = 3 Vertices: (4,0) and (-4,0) c2 = a2 + b2 c2 = 16 + 9 c2 = 25 c = 5 Foci: (5,0) and (-5,0)

8. Vertices: (0,4) and (0,-4) Foci: (0,5) and (0,-5) a2 = 16 a = 4 b2 = 9 b = 3 c2 = a2 + b2 c2 = 16 + 9 c2 = 25 c = 5 Asymptotes: Domain = R Range = -,-4]  [4, 

9. Test (0,0) 0 < 1 True

10. le fin КОНЕЦ τέλοσ final The end finito ﭙﺎﻴﺎﻥ sof