In the previous section, we completed a discussion on ellipse. In this section, we will see hyperbola.
Some basics about hyperbola can be written in 6 steps:
1. Consider the five points F1, F2, P1, P2 and P3 marked in fig.11.42 below:
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| Fig.11.42 |
2. Let us write some distances:
• Distance of P1:
♦ Distance of P1 from F1 is 8.6 units.
♦ Distance of P1 from F2 is 18.7 units.
✰ The difference of the two distances is (18.7 - 8.6) = 10.1 units.
• Distance of P2:
♦ Distance of P2 from F1 is 16.7 units.
♦ Distance of P2 from F2 is 6.6 units.
✰ The difference of the two distances is (16.7 - 6.6) = 10.1 units.
• Distance of P3:
♦ Distance of P3 from F1 is 14.1 units.
♦ Distance of P3 from F2 is 4.0 units.
✰ The difference of the two distances is (14.1 - 4.0) = 10.1 units.
3. So the three points P1, P2 and P3 have a specialty. It can be written in two steps:
(i) Take any one of those three points.
♦ Measure the distance of that point from F1.
♦ Measure the distance of that point from F2.
(ii) The difference of the two distances will be 10.1 units.
4. There are infinite number of points for which the difference is 10.1 units.
• All such points will lie in the red curve.
• The red curve is called a hyperbola.
5. So we can write the definition:
A hyperbola is the set of all points in a plane difference of whose distances from two fixed points in the plane is a constant.
6. Note that, for a particular hyperbola, the two points F1 and F2 are fixed. If we change one or both of those points, we will get another hyperbola.
Now we will see some basic features of hyperbola. They can be written in 8 steps:
1. We have seen that, F1 and F2 are two fixed points.
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They are called the foci of the hyperbola.
(‘foci’ is the plural of ‘focus’)
2. Draw a line connecting the two foci.
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Let it intersect the hyperbola at A and B.
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Then the line through the foci is called the transverse axis of the hyperbola.
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This is shown in fig.11.43 below:
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| Fig.11.43 |
3. The midpoint of F1F2 is called center of the hyperbola. It is denoted by the letter O.
4. Draw a line through O and perpendicular to AB.
• This line is called the conjugate axis of the hyperbola.
5. The points A and B at which the transverse axis meets the hyperbola are called vertices of the hyperbola.
6. The length AB is written as ‘2a’. This is shown in fig.11.44 below:
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| Fig.11.44 |
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The distance between the two foci is written as ‘2c’
7. Based on the above fig.11.44, we can write:
♦ Distance from center to any one vertex is 'a'.
♦ '2a' is the length of the transverse axis.
♦ Distance from center to any one focus is 'c'.
8. '2b' is considered as the length of the conjugate axis.
♦ Where $b=\sqrt{c^2 - a^2}$
In fig.11.42, we saw that, the difference of the distances is a constant. Our next aim is to find the value of that constant. It can be written in 3 steps:
1. Consider a point P on the hyperbola.
• Let it be situated at the vertex A
• Then the distance of P from F1 = AF1 = (OF1 – OA) = c-a
• Also the distance of P from F2 = AF2 = (OA + OB + BF2)
= [a+a+(c-a)] = c + a
2. So the difference between the distances = [(c+a) – (c-a)] = [c+a-c+a] = 2a
3. We can write:
For any point P on the hyperbola, the difference in distances from the foci will be ‘2a’
Eccentricity of a Hyperbola
This can be explained in 5 steps:
1. Consider the distance ‘c’.
• It is the distance of the focus from the center O.
2. Consider the distance ‘a’.
• It is the distance of the vertex from the center O.
3. Eccentricity of a hyperbola is the ratio of the above two items. It is denoted by the letter ‘e’.
So we can write: $\rm{e=\frac{c}{a}}$
4. Based on this result, we can write: c = ae.
• That means, in any hyperbola, the focus is at a distance of ae from the center.
5. From fig.11.44, it is clear that, 'c' is greater than 'a'. So the eccentricity is never less than 1.
In the next section, we will see the standard equations of a hyperbola.
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